Fakultät Mathematik und Naturwissenschaften Fachrichtung Physik, Institut für Kern- und Teilchenphysik

Professur für Phänomenologie der Elementarteilchen

PHENOMENOLOGICAL STUDY OF THEMINIMAL R-SYMMETRICSUPERSYMMETRIC STANDARDMODEL

Dipl. Phys. Philip DießnerGeboren am: 8. Dezember 1986 in Schmalkalden

DISSERTATION

zur Erlangung des akademischen Grades

DOCTOR RERUM NATURALIUM (DR. RER. NAT.)

Erstgutachter

Prof. Dr. Dominik Stöckinger

Zweitgutachter

Prof. Dr. Jan KalinowskiBetreuender Hochschullehrer

Prof. Dr. Dominik Stöckinger

Eingereicht am: 9. August 2016Verteidigt am: 20. Oktober 2016

ABSTRACT

The Standard Model (SM) of particle physics gives a comprehensive description of numer-ous phenomena concerning the fundamental components of nature. Still, open questionsand a clouded understanding of the underlying structure remain. Supersymmetry is awell motivated extension that may account for the observed density of dark matter in theuniverse and solve the hierarchy problem of the SM.

The minimal supersymmetric extension of the SM (MSSM) provides solutions to thesechallenges. Furthermore, it predicts new particles in reach of current experiments. However,the model has its own theoretical challenges and is under fire from measurements providedby the Large Hadron Collider (LHC).

Nevertheless, the concept of supersymmetry has an elegance which not only shines inthe MSSM. Hence, it is also of interest to examine non-minimal supersymmetric models.They have benefits similar to the MSSM and may solve its shortcomings. R-symmetry is theonly global symmetry allowed that does not commutate with supersymmetry and Lorentzsymmetry. Thus, extending a supersymmetric model with R-symmetry is a theoreticallywell motivated endeavor to achieve the complete symmetry content of a field theory. Sucha model provides a natural explanation for non-discovery in the early runs of the LHC andleads to further predictions distinct from those of the MSSM.

The work described in this thesis contributes to the effort by studying the minimalR-symmetric supersymmetric extension of the SM (MRSSM). Important aspects of itsphysics and the dependence of observables on the parameter space of the MRSSM areinvestigated.

The discovery of a scalar particle compatible with the Higgs boson of the SM at the LHCwas announced in 2012. It is the first and crucial task of this thesis to understand theunderlying mechanisms leading to the correct Higgs boson mass prediction in the MRSSM.Then, the relevant regions of parameter space are investigated and it is shown that theyare also in agreement with other Higgs observables. Another observable that is measuredwith great accuracy and especially sensitive to corrections from additional supersymmetricstates is the mass of the W boson. Contributing effects within the MRSSM are identifiedand their dependency on the model parameters is studied.

The presence of a stable supersymmetric particle as candidate for dark matter is aprediction of the MRSSM. The interplay of the relevant processes generating the correctabundance of dark matter in the universe and explaining the non-discovery by directsearches is investigated. Moreover, results of Run 1 of the LHC are used to study theelectroweak MRSSM sector. This leads to a classification of viable regions of parameterspace consistent with dark matter and LHC constraints.

In the last part of this thesis the different observables are analyzed in coherence. Thisallows to identify valid regions of parameter space and highlights promising predictions ofthe MRSSM for the coming runs of the LHC and other experiments.

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KURZDARSTELLUNG

Das Standardmodell (SM) der Elementarteilchenphysik liefert eine prägnante Beschreibungder Phänomene, welche die grundlegenden Bestandteile der Natur betreffen. Es verbleibenaber weiterhin offene Fragen und eine fehlende Einsicht in die zugrunde liegenden Struk-turen. Supersymmetrie ist eine wohl begründete Erweiterung, welche es ermöglicht diebeobachtete dunkle Materiedichte im Universum zu erklären und das Hierarchieproblemdes SM zu lösen.

Die minimale supersymmetrische Erweiterung des SM (MSSM) besitzt diese Eigen-schaften. Darüber hinaus sagt es neue Teilchen in Reichweite aktueller Experimente vorher.Die eigenen theoretischen Herausforderungen des Modells und Einschränkungen durchMessungen am Large Hadron Collider (LHC) schränken es jedoch stark ein.

Dennoch birgt das Konzept der Supersymmetrie eine Eleganz, die eine ansprechendeGrundlage für weitere Modelle bietet. Daher ist es auch von Interesse, nicht-minimalesupersymmetrische Modelle zu untersuchen. Diese bieten mit dem MSSM vergleichbareVorteile und können dessen Diskrepanzen auflösen. R-Symmetrie ist die einzig möglicheglobale Symmetrie, die nicht mit Super- und Lorentzsymmetrie kommutieren. Ein auf dieseWeise konstruiertes Modell enthält somit alle grundlegenden Symmetrien einer Feldtheorie.Durch die Inklusion von R-Symmetrie können die bisherige Nichtentdeckung am LHC erklärtund vom MSSM unterscheidbare Vorhersagen gemacht werden.

In dieser Arbeit wird die Untersuchung des minimale R-symmetrische supersymme-trische Erweiterung des SM (MRSSM). Wichtige Aspekte der Phänomenologie und dieAbhängigkeit der Observablen von den Parametern des MRSSM werden untersucht.

Die Entdeckung eines skalaren Teilchens kompatibel mit dem Higgs-Boson des SM amLHC wurde im Jahre 2012 bekannt gegeben. Die Untersuchung der zugrunde liegendenMechanismen, welche die Masse des Higgs Bosons im MRSSM korrekt verwirklichen,ist Hauptbestandteil des erste Teils dieser Arbeit. Dabei wird der Parameterraum desModells untersucht und gezeigt, dass auch Übereinstimmung mit weiteren Observablender Higgsphysik möglich ist. Ein weitere wichtige Messgröße, welche mit hoher Genauigkeitbestimmt und empfindlich auf Beiträge supersymmetrischer Teilchen ist, ist die Masse desW Bosons. Beiträge innerhalb des MRSSM werden identifiziert und ihre Abhängigkeit vonModellparametern untersucht.

Die Existenz eines stabilen supersymmetrischen Teilchens als Kandidat für dunkle Ma-terie ist eine Vorhersage des MRSSM. Es wird untersucht, wie die relevanten Prozessezusammenspielen, um die korrekte Dichte an dunkler Materie im Universum zu erzeugenund die Nichtentdeckung bei direkte Suche zu erklären. Des weiteren werden die ers-ten Ergebnisse des LHC verwendet, um den elektroschwachen Sektor des MRSSM zuuntersuchen.

Im letzten Teil dieser Arbeit wird das Zusammenspiel verschiedener Observablen analy-siert. Auf diese Weise können erlaubte Parameterregionen festgestellt und Vorhersagenfür zukünftige Experimente gemacht werden.

iv

CONTENTS

List of Figures ix

List of Tables xiii

1. Introduction 1

2. Supersymmetry 32.1. The Standard Model and beyond . . . . . . . . . . . . . . . . . . . . . . . . 32.2. General considerations on supersymmetry . . . . . . . . . . . . . . . . . . 4

2.2.1. Super–Poincaré algebra . . . . . . . . . . . . . . . . . . . . . . . . . 52.2.2. Superspace and superfields . . . . . . . . . . . . . . . . . . . . . . . 52.2.3. Supersymmetric Lagrangian . . . . . . . . . . . . . . . . . . . . . . . 72.2.4. SUSY breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3. R-symmetry for model building . . . . . . . . . . . . . . . . . . . . . . . . . 122.4. The Minimal Supersymmetric Standard Model . . . . . . . . . . . . . . . . 14

2.4.1. Symmetries and fields . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4.2. SUSY breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4.3. Electroweak symmetry breaking . . . . . . . . . . . . . . . . . . . . 16

2.5. R-symmetric Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 162.5.1. Symmetries and fields . . . . . . . . . . . . . . . . . . . . . . . . . . 172.5.2. SUSY breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.5.3. Electroweak symmetry breaking . . . . . . . . . . . . . . . . . . . . 202.5.4. Symmetry transformations of the MRSSM Lagrangian . . . . . . . . 20

2.6. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3. Input parameters for the MRSSM 233.1. Field theoretical concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1.1. Regularization and renormalization . . . . . . . . . . . . . . . . . . . 233.1.2. Effective action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.1.3. Effective potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.1.4. Renormalization group equations . . . . . . . . . . . . . . . . . . . . 283.1.5. Effective field theories . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2. Application to the MRSSM . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2.1. Renormalization of the MRSSM . . . . . . . . . . . . . . . . . . . . 29

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Contents

3.2.2. Matching to the SM . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.2.3. Pole mass prediction for SUSY states . . . . . . . . . . . . . . . . . 31

3.3. Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.3.1. Comparison to experiment . . . . . . . . . . . . . . . . . . . . . . . 32

3.4. Scenarios in the MRSSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.4.1. Normal ordered scenario . . . . . . . . . . . . . . . . . . . . . . . . 333.4.2. Light singlet scenario . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.5. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4. SUSY Higgs physics 374.1. The Higgs sector of the MRSSM . . . . . . . . . . . . . . . . . . . . . . . . 384.2. Quantum corrections to the CP-even Higgs boson mass . . . . . . . . . . . 39

4.2.1. Diagrammatic approach . . . . . . . . . . . . . . . . . . . . . . . . . 394.2.2. Effective potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.2.3. One-loop corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.2.4. Two-loop corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.2.5. Higher-order uncertainties and scheme dependency . . . . . . . . . 47

4.3. Higgs observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.4. Phenomenology of a light singlet scenario . . . . . . . . . . . . . . . . . . . 50

4.4.1. Mixing of states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.4.2. Higgs observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.4.3. Mass contribution from mixing . . . . . . . . . . . . . . . . . . . . . 54

4.5. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5. Electroweak precision observables 575.1. Electroweak mixing angle and mass of the W boson . . . . . . . . . . . . . 57

5.1.1. Formal one-loop corrections . . . . . . . . . . . . . . . . . . . . . . . 585.1.2. Higher-order QCD corrections . . . . . . . . . . . . . . . . . . . . . . 595.1.3. Theoretical uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.2. Electroweak precision observables . . . . . . . . . . . . . . . . . . . . . . . 615.2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.2.2. Qualitative discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.3. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6. Dark matter in the MRSSM 696.1. The MRSSM and dark matter . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6.1.1. Dark matter scenarios of the MRSSM . . . . . . . . . . . . . . . . . 706.1.2. Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6.2. Dark matter relic density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716.2.1. Processes in the MRSSM . . . . . . . . . . . . . . . . . . . . . . . . 726.2.2. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.3. Direct detection limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 766.3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 766.3.2. Processes in the MRSSM . . . . . . . . . . . . . . . . . . . . . . . . 786.3.3. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6.4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

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7. Direct SUSY searches at the LHC 837.1. SUSY phenomenology at a particle collider . . . . . . . . . . . . . . . . . . 83

7.1.1. The LHC and experiment description . . . . . . . . . . . . . . . . . . 847.1.2. Experimental SUSY searches . . . . . . . . . . . . . . . . . . . . . . 85

7.2. The MRSSM at the LHC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877.2.1. MRSSM signatures . . . . . . . . . . . . . . . . . . . . . . . . . . . 877.2.2. Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

7.3. Electroweak LHC searches in the MRSSM . . . . . . . . . . . . . . . . . . . 907.3.1. Discriminating the MRSSM from the MSSM . . . . . . . . . . . . . 917.3.2. Sleptons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 947.3.3. Charginos and neutralinos . . . . . . . . . . . . . . . . . . . . . . . . 95

7.4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

8. Exploring the parameter space 1018.1. Constraining the normal ordered scenario . . . . . . . . . . . . . . . . . . . 1018.2. Constraining the light singlet scenario . . . . . . . . . . . . . . . . . . . . . 1078.3. The normal ordered scenario and dark matter . . . . . . . . . . . . . . . . . 1098.4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

9. Conclusions 115

A. Tools and Input parameters 117A.1. Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117A.2. Tool overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

A.2.1. MicrOMEGAs and LUXcalc interface . . . . . . . . . . . . . . . . . . . 118A.3. Numerical values of SM input parameters . . . . . . . . . . . . . . . . . . . 119

B. MRSSM renormalization group functions 121B.1. Beta functions of gauge couplings . . . . . . . . . . . . . . . . . . . . . . . 121B.2. Beta functions of superpotential parameter . . . . . . . . . . . . . . . . . . 121

B.2.1. Trilinear parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 121B.2.2. Bilinear parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

B.3. Beta functions of soft breaking parameters . . . . . . . . . . . . . . . . . . 124B.3.1. Bilinear parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124B.3.2. Dirac mass parameters . . . . . . . . . . . . . . . . . . . . . . . . . 124B.3.3. Scalar soft breaking mass parameters . . . . . . . . . . . . . . . . . 124

C. The MRSSM after electroweak symmetry breaking 131C.1. Scalar potential and tadpole equations . . . . . . . . . . . . . . . . . . . . . 131C.2. Masses of the Higgs fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 132C.3. Mass matrices of R-Higgses . . . . . . . . . . . . . . . . . . . . . . . . . . 133C.4. Mass matrices of squarks and sleptons . . . . . . . . . . . . . . . . . . . . 133C.5. Mass matrices of neutralinos and charginos . . . . . . . . . . . . . . . . . . 135

D. Exotic decay patterns of states for the light singlet scenario 137

Bibliography 141

vii

LIST OF FIGURES

3.1. Shown are the mass spectrum plots of the Benchmark points. The leftpanels give the spectra for normal ordered BMP1 to BMP3 from top tobottom. The right plots give them for the light singlet BMP4 to BMP6 fromtop to bottom. The plots were generated using PySLHA [147]. . . . . . . . . 36

4.1. Generic scalar self energies . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2. For varying each of the four superpotential parameters λd , λu, Λd and Λu

three panels are shown. All other parameters are chosen as for BMP1. Thetop plot contains the prediction of the SM-like Higgs boson mass at tree-level(blue, dashed-dotted), one-loop (green, dashed) and two-loop (red, full) level,the middle one the difference of the full one-loop Feynman diagrammaticand effective potential approach and the bottom panel the difference ofthe two-loop and full one-loop result. The lines are not continued when atachyonic state is encountered at tree-level mass calculation. . . . . . . . . 43

4.3. Shown are the same plots as in fig. 4.2 with the unvaried parameters chosenas for BMP2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.4. Shown are diagrams contributing to the two-loop effective potential via thestrong coupling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.5. The figure compiles the numeric investigation of two-loop contributions tothe mass of the SM-like Higgs boson depending on MD

O. The left plot showsthe contributions from the different sectors, all other parameters are setas BMP3. The middle (BMP1) and right (BMP3) plot give for the MRSSMtwo lines for scenarios with different sgluon masses and without sgluoncontributions as well as for the MSSM lines for comparable scenarios withand without stop mixing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.6. The exclusion limit by HiggsBounds is given in the mA-tanβ parameter plane.For each point Λu has been varied to achieve the correct SM-like Higgs bosonmass. The yellow region is allowed, while the blue region is excluded byHiggsBounds. All other parameters are chosen as for BMP3. . . . . . . . . 49

ix

List of Figures

4.7. Shown is the squared mixing matrix element (ZH12)2 of the Higgs mass

matrix (C.6) depending on the lightest Higgs boson mass for a scan overparameter points. All points of the following fig 4.8 are included. The markerand its color show if it is allowed (yellow circle), excluded by HiggsBounds(blue down-triangles), by HiggsSignals (blue up-triangles) or by both pro-grams (violet diamonds). The upper left of the plot is not populated by pointsas the SM-like Higgs boson mass has been fixed to 125 GeV and it is notpossible to achieve strong mixing for lower masses of the light state. . . . 52

4.8. The exclusion limits by HiggsBounds and HiggsSignals are given in themS-MD

B parameter plane. For each point Λu has been varied to achieve thecorrect SM-like Higgs boson mass. The yellow region is allowed, while theblue (green) region is excluded by HiggsBounds ( HiggsSignals) and theviolet region by both tools. For the left plot λu = 0.0, while it is −0.01 for theright plot. All other parameters are chosen as for BMP6. . . . . . . . . . . . 53

4.9. Shown are plots for mixing of the singlet and doublet state in the MRSSMdepending on MD

B (left) and m2S (right). The top panels show the masses of

the singlet-like (dashed, blue) and SM-like state (full, green). The secondrow gives the mixing matrix elements for the mix-in of the singlet (up-typedoublet) into the lightest state as dashed blue (full green) line. In the bottomrow the difference of the SM-like Higgs boson mass to the mass, when MD

Band m2

S are of a magnitude as in the normal ordered scenario, is shown. Allnon-varied parameters are chosen as for BMP4. . . . . . . . . . . . . . . . 55

5.1. Shown are Feynman diagrams depicting generic gauge boson self-energycontributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.2. The panels show the comparison of the electroweak precision observablesdepending on Λu, calculated using the approximations given by eqs. (5.31) forthe chargino and neutralino contributions. All other parameters are chosen asfor benchmark point 3 (left, normal ordered scenario) or 5 (right, light singletscenario). The plot for BMP5 does not extend beyond 1.5 as tachyonic statesappear in this range. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.3. The plots show the comparison of the mass of the W boson dependingon Λu, calculated using full MRSSM contributions and different approxima-tions for the electroweak precision parameters: neutralino and charginosector, eqs. (5.31) and (5.32), Higgs sector, eqs. (5.29), and R-Higgs sector,eqs. (5.30), as well as the stop contributions of eqs. (5.28), and the tree-levelcontribution from the triplet vev. All other parameters are chosen as forbenchmark point 3 (left, normal ordered scenario) or 5 (right, light singletscenario). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6.1. Depicted are Feynman diagrams relevant for the generation of the darkmatter relic density via annihilation through stau or Z boson. . . . . . . . . . 73

6.2. Shown is the dark matter relic density depending on the LSP and right-handedstau mass. The left panel highlights the light singlet scenario, with all con-stant parameters chosen as for BMP4. The normal ordered scenario is thebasis for the right panel, where all constant parameters are chosen as forBMP3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

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List of Figures

6.3. Shown is the dark matter relic density depending on the LSP and right-handedstau mass. All other slepton masses are set to the varied stau mass, theremainder of the parameters are chosen as in BMP3. The left panel doesnot contain coannihilation effects, the right panel does. . . . . . . . . . . . 75

6.4. Shown is the dark matter relic density depending on the LSP mass andthe mass splitting between the LSP and the lightest chargino. All otherparameters are chosen as for BMP2. . . . . . . . . . . . . . . . . . . . . . . 76

6.5. Shown is an overview of the direct detection exclusion limits from differentcollaborations. Taken from ref. [234]. . . . . . . . . . . . . . . . . . . . . . 77

6.6. Shown are the Feynman diagrams relevant for dark matter direct detectionexperiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.7. Shown are the 90% and 95% C.L. direct detection exclusion contours(marked by the corresponding p-value) in the µu-msquark parameter plane.The black line marks the of approximate complete destructive interferencebetween the different processes (6.17) as described in the text. The left(right) plot exemplifies the normal ordered (light singlet) scenario as allunvaried parameters are chosen as for BMP3 (BMP6). . . . . . . . . . . . . 81

7.1. Illustration of the production of electroweak SUSY states at the LHC. Theincoming protons provide the necessary quark and anti-quark. These interactelectroweakly to produce a pair of charginos and neutralinos or a pair ofsleptons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

7.2. Relevant Feynman diagrams for the decay of the electroweak SUSY states. 917.3. Exclusion plots in the chargino-neutralino and stau mass plane for the first

scenario of tab. 7.4 in the MSSM (left) and in the MRSSM (right). The violet(blue) region marks the 90% (95%) C.L. exclusion limits given by CheckMATE,each denoted by 1 − C.L.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

7.4. Exclusion plots in the LSP and slepton mass plane for right-handed (left) andleft-handed (right) selectron and smuon. Taken from ATLAS ref. [246], fig.8(a) and 8(b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

7.5. Exclusion plots in the LSP and slepton mass plane for right-handed (left)and left-handed (right) selectron and smuon. The violet (blue) region marksthe 90% (95%) C.L. exclusion limits given by CheckMATE, each denoted by1 − C.L.. All other unvaried parameters as for BMP2. . . . . . . . . . . . . . 96

7.6. Exclusion limits in the MRSSM for the chargino and neutralino productionassuming heavy sleptons as a function of the higgsino and wino-triplinomasses (left) and of the two higgsino masses (right). The violet (blue)region marks the 90% (95%) C.L. exclusion limits given by CheckMATE, eachdenoted by 1 − C.L.. All parameters which are not varied in the plots arefixed to the values of BMP5. . . . . . . . . . . . . . . . . . . . . . . . . . . 96

7.7. Exclusion limits in the MRSSM for the chargino and neutralino productionassuming light staus as a function of the two higgsino mass parameters µd,u.The violet (blue) region marks the 90% (95%) C.L. exclusion limits given byCheckMATE, each denoted by 1 − C.L.. The left plot has light right-handedslepton masses of 100 GeV. In the right plot only the right-handed stau massis fixed to 100 GeV. All other parameters are set to the values of BMP5.Most importantly, the Dirac wino mass is set to MD

W = 500 GeV. . . . . . . 97

xi

List of Figures

8.1. Shown are the SM-like Higgs boson mass, given by the color contours, andthe W boson mass, given by the labeled black contour lines for differentcombinations of superpotential couplings. The other parameters are set toBMP1, BMP2 and BMP3 for the top, middle and bottom row, respectively.The benchmark point is marked by a star in each plot. . . . . . . . . . . . . 104

8.2. Shown are the SM-like Higgs boson mass, given by the color contours, andthe W boson mass, given by the labeled black contour lines for differentcombinations of µu and Dirac mass parameters. The selection of benchmarkpoints is as in fig. 8.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

8.3. Shown are the SM-like Higgs boson mass, given by the color contours, andthe W boson mass, given by the labeled black contour lines for differentcombinations of Dirac mass parameter and superpotential couplings. Theselection of benchmark points is as in fig. 8.1. . . . . . . . . . . . . . . . . . 106

8.4. Shown are the SM-like Higgs boson mass (color contours), the W bosonmass (black full lines), the dark matter relic density (blue dashed lines) andthe regions excluded by dark matter direct searches (violet region), wherethe red line marks the 95% C.L. exclusion. For the top row and the left panelof the bottom row all other parameters are set to the BMP5 values, for theright plot on the bottom row to the ones of BMP4. . . . . . . . . . . . . . . 108

8.5. Shown are the same observables as in fig.8.4. The unvaried parameters areset to the benchmark points BMP7 (top, left), BMP8 (top, right) and BMP9(bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

xii

LIST OF TABLES

2.1. R-charges of different superspace objects appearing in the text. The last twoare the chiral and vector superfield with their respective component fields. 8

2.2. Shown are the superfields of the MSSM and their respective componentfields as well as their group representation under SU(3)C × SU(2)L × U(1)Y .For U(1)Y the quantum number Y/ 2 is given. The index i ∈ 1, 2, 3 runs overthe three generations of the matter fields. The charge conjugated spinorfield ψC is derived from ψ via the relation ψC = C ψ

Twith C = iγ2γ0. . . . . 14

2.3. Shown are the additionally introduced chiral superfields of the MRSSM andtheir respective component fields as well as their group representationunder SU(3)C × SU(2)L × U(1)Y . For U(1)Y the quantum number Y/ 2 is given.It should be noted that the two components of the triplet charged underU(1)em are independent as T *

− 6= T+. . . . . . . . . . . . . . . . . . . . . . . 17

2.4. The R-charges of the superfields and the corresponding bosonic and fermioniccomponents of the MRSSM are collected in this table. . . . . . . . . . . . 17

3.1. Benchmark points of the normal ordered scenario. Dimensionful parametersare given in GeV or GeV2, as appropriate. The first part gives input parame-ters, with values specific for each point, the second, where the values areset to a common number for all points. The last part shows the derivedparameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2. Benchmark points for the light singlet scenario. Layout is as for tab. 3.1. . . 35

4.1. Definition of the fixed parameters for the MSSM points in fig. 4.5. Allparameters in GeV or GeV2, where appropriate. The stop mixing parameterXt = At − µ cotβ is given both for the case of no and large stop mixing. . . 47

6.1. Initial and respective final states for the coannihilation of the bino-singlinowith the sleptons. For the last two rows the charge conjugate states arealso taken into account. ` ∈ e,µ, τ . . . . . . . . . . . . . . . . . . . . . . . 73

6.2. Initial and respective final states for the coannihilation of the bino-singlinowith the wino-triplino. For the last three rows the charge conjugate statesare also taken into account. q ∈ u, c, t, q′ ∈ d, s, b, ` ∈ e,µ, τ . . . . . . 74

xiii

List of Tables

7.1. Electroweak cross sections at a center-of-mass energy of√

s = 8 TeV ascalculated by Herwig++ using the SARAH model input. All values given in fb.The cross sections are calculated at leading order with uncertainties of theorder of ten percent determined from scale variation. The integration error isbelow one percent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

7.2. Electroweak cross sections at a center-of-mass energy of√

s = 8 TeV ascalculated by MadGraph using the SARAH model input. All values given in fb.The cross sections are calculated at leading order with uncertainties of theorder of ten percent determined from scale variation. The integration error isbelow one percent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

7.3. Masses of the non-SM particles in the BMPs relevant for the LHC studiesdiscussed here. All values given in GeV. . . . . . . . . . . . . . . . . . . . . 91

7.4. Possible discovery scenarios at colliders of different sets of particles fromthe neutralino-chargino sector and the corresponding dominant gauge eigen-states for the MSSM and MRSSM. This is assuming the light singlet scenarioin the MRSSM, which leads to a light bino-singlino as LSP candidate. . . . . 93

8.1. Shown are the changes to the benchmark points of the normal orderedscenario from tab. 3.1 to arrive at the benchmark points in agreement withall considered observables. Dimensionful parameters are given in GeV orGeV2, as appropriate. The last part shows the derived parameters. . . . . . 110

8.2. Given are the relevant observables for all benchmark points defined in thisthesis. The first part gives the important masses including those neces-sary for dark matter constraints, In the middle section the passing of allof the HiggsBounds limits is denoted and the value for the relic density isgiven. The last three rows give the p-values for the Higgs observables fromHiggsSignals and respective direct searches, DM and LHC (actually CLs). 114

xiv

1. INTRODUCTION

The ambition of elementary particle physics is to describe the fundamental building blocksof the universe and their respective interactions contained by an overarching theory. TheStandard Model of particle physics (SM) is the most compact and accurate theory offundamental forces known to date. The SM contains all known elementary particles andtheir respective interactions. The mechanism of spontaneous symmetry breaking is usedto generate masses for the fields, and a remnant of the Higgs mechanism is the predictionof a fundamental scalar particle, the Higgs boson.* Only in 2012, a valid candidate for thisstate has been found at the Large Hadron Collider (LHC).

With the discovery of the Higgs boson the last missing piece of the SM is unveiled.However, it is also clear that the SM can not be the final description of nature. The SMfaces experimental and theoretical challenges which will be addressed at the TeV scale.

From astrophysical evidence it is known that around 80% of the universe’s matter contentdoes not interact electromagnetically. One interpretation is that this is made up of unknownelementary particles, commonly called dark matter. There is no viable candidate in the SMfor dark matter, and even as it has not been detected directly, an extension of the SM isrequired to explain the astrophysical observations.

The Higgs boson is the only fundamental scalar in the SM. Its mass is not protectedby any symmetry allowing for quantum corrections from all scales to affect it. Gravitybecomes important for particle physics at the Planck scale.† If there is no new physics up tothis scale it can be expected that the mass of the Higgs boson receives radiative quantumcontributions of the order of the Planck scale and therefore should be in magnitude close toit. The measured value is below the TeV scale and this discrepancy between the theoreticalexpectation and experimental result is known as the hierarchy problem of the SM.

Supersymmetry (SUSY) is a well motivated extension of the SM solving some of itsproblems. SUSY provides a connection between the bosons and the fermions and leadsto the same number of bosonic and fermionic degrees of freedom in a theory. ImposingSUSY onto the SM then requires additional fields in the model to achieve this equilibrium.The hierarchy problem of the SM is solved by SUSY with the addition of new quantumcorrections to the Higgs boson mass. The contributions from fermionic and bosonic statescombine in such a way that the scale dependency is diminished.

*This is the usually utilized short form for Brout–Englert–Guralnik–Hagen–Higgs–Kibble mechanism.†Experiments with energies in the TeV range probe distances as small as 10−17 cm. Gravity becomes relevant

at distances of 10−33 cm or energies of 1015 TeV, known as the Planck scale.

1

1. Introduction

To explain dark matter in the context of a supersymmetric model an additional symmetryneeds to be imposed. In the minimal SUSY extension of the SM (MSSM) this is a discretesymmetry separating the SM fields from the new states. Another possibility is theintroduction of R-symmetry. It is the only allowed continuous symmetry that commuteswith SUSY and distinguishes between the known SM fields and their supersymmetricpartners. The inclusion of R-symmetry maximizes the symmetry content of SUSY model.R-symmetry is incorporated in the Minimal R-symmetric SUSY Standard Model (MRSSM)and leads to a dark matter candidate being realized in the model.

The MSSM is a well established model and its phenomenology has been studiedextensively in the past. This is not true for the MRSSM. Features of the model have beenstudied before individually, but not the interplay of several distinct aspects. This is thegoal of this thesis. This avenue of research is of interest as the MRSSM makes novelpredictions. For example, the additional fermionic states are not of Majorana nature as inthe MSSM but rather Dirac type. A Dirac fermion, unlike Majorana states, is not its ownantiparticle. Hence, the number of degrees of freedom for certain SUSY states is doubled,when comparing both models. This leads to distinct experimental signatures and an uniqueconnection between the bosonic and fermionic sector of the model.

This thesis is structured as follows: In chapter 2 the concepts of SUSY and R-symmetryare introduced, and it is described how they can be used to extend the SM solving theaforementioned problems introducing the MRSSM. The technical details are described inchapter 3. It is shown how the MRSSM is matched to experiment and how a consistentquantum field theoretical calculation is carried out leading to meaningful predictions. Finally,two relevant scenarios are identified which illustrate regions in parameter space with aninteresting phenomenology.

One of the foremost predictions of every realistic SUSY model is the mass of the Higgsboson. It is a free parameter in the SM but not in the MSSM or MRSSM. As the candidatefor this state has been recently discovered at the LHC, it is of prime interest to understandwhether or not the prediction of the MRSSM can be brought in agreement with theexperimental measurement. This is done in chapter 4. There, the underlying mechanismto generate the mass in the MRSSM is analyzed. Moreover, a study of further Higgsobservables is performed to understand singular properties of the benchmark scenarios.

Even as introducing SUSY solves several of the SM’s problems, every model faces thechallenge to describe all previous experimental observations to the same degree as theSM. This is the focus of chapter 5. There, the mass of the W boson is used to ensurethat additional MRSSM contributions to electroweak precision observables stay within theexperimental uncertainties.

In chapter 6 the lightest neutralino is identified as a valid dark matter candidate of theMRSSM. The precisely measured dark matter relic density and results from direct searchesare used to identify the relevant processes of (co)annihilation and the mass range of thedark matter candidate.

The requirement of SUSY introduces a multitude of additional states and, to solve thehierarchy problem of the SM, several of them should be within reach of the LHC. Inchapter 7 the results of Run 1 of the LHC are used to investigate the electroweak MRSSMsector. Differences in the LHC phenomenology of the MSSM and MRSSM are identified.

The results from all the studies are combined in chapter 8, and the correlations ofthe different observables are analyzed. Understanding of the interplay of all states andparameters in model allows then to identify the viable parameter regions of the MRSSM.

2

2. SUPERSYMMETRY

Supersymmetry (SUSY) at the TeV scale is a well motivated extension of the StandardModel of particle physics (SM). It solves several of the SM’s shortcomings. For example, itprovides a solution to the hierarchy problem, and a SUSY model can include a candidate fordark matter. With an extended particle content it gives rise to an interesting phenomenologyaccessible by experiment. In this chapter a comprehensive introduction to supersymmetryis given highlighting the relevance of R-symmetry, which is the central topic of this thesis.

2.1. THE STANDARD MODEL AND BEYOND

Studying supersymmetry at the TeV scale is not possible without knowing about its lowenergy limit, the SM. Hence, a short summary of the SM is given here and certain relationsof it are highlighted through this work when recovered from a supersymmetric model.

The SM contains all known elementary particles and describes all known non-gravitationalforces on a quantum level. Quantum Chromodynamics (QCD) gives rise to the strongforce while electromagnetism and the weak force are unified in the electroweak forceand masses for the elementary particles are generated using spontaneous electroweaksymmetry breaking.*

The SM is a quantum field theory and, following Yang and Mills [1], has internal sym-metries described by a non-simple Lie group. This gauge group is the well-knownSU(3)C × SU(2)L × U(1)Y , where the indices stand for color (C), left-handed chirality (L)and hypercharge (Y). Elementary particles are described by fields of the theory and theirexcitations, each in a certain representation of the Lie and the Poincaré group. There is avector field in the adjoint representation for each simple sub-group. The fermionic matterfields are known as leptons ` and quarks q, transforming trivially or non-trivially under theSU(3)C, respectively. The scalar SU(2)L Higgs doublet gives rise to electroweak symmetrybreaking [2–7]. This yields massive vector bosons Zµ, W±

µ , the massive scalar Higgs bosonh0 and the massless photon Aµ of quantum electrodynamics [8]. The gluon gµ remainsmassless as QCD is not affected by the symmetry breaking. The masses of the matterfields originate from their Yukawa couplings to the Higgs doublet.

*In this thesis, neutrinos are assumed to be massless as their masses are so far observed only indirectlyvia the phenomenon of neutrino oscillations and their nature as either Dirac or Majorana fermions is yetunknown. Either way, these masses give rise to an extension of the SM as we know it. This is rather aproblem at the low energy frontier and not in the focus of this work.

3

2. Supersymmetry

Over the decades, numerous predictions of the SM have been verified experimentally andthe SM is consistent with many measurements. Still, there are a number of observationswhich can not be explained. In the following, some points are given where supersymmetryprovides solutions to some of the SM’s problems.

Amongst other, supersymmetric models can provide a candidate for dark matter [9,10]. Dark matter is the dominant matter component of the known universe that does notinteract electromagnetically and for which astrophysical evidence exists [11]. Dark mattercan be provided in a SUSY model if there is a new particle that is stable and has no electriccharge. An example of another observable where the SM prediction is in conflict with theexperiment measurement is the anomalous magnetic moment of the muon. In a SUSYmodel additional contributions exist so that agreement is possible [12, 13].

On the theoretical side, SUSY at the TeV scale may give a solution to the hierarchyproblem [14–17], asking why the weak force is much stronger than the gravitational one.Phrased as a fine-tuning problem, it is not clear why the Higgs boson mass is so smallcompared to the Planck scale. As the Higgs boson is a scalar and its mass is not protectedby any symmetry, the mass receives corrections that generally depend quadratically ona new scale where physics beyond the Standard Model becomes important. Withoutother extensions to the SM this scale is the Planck scale MP = O(1019 GeV) when thegravitational force becomes comparable to the other forces and large cancellations need tohappen to tune the mass to the electroweak scale v = O(102 GeV) as it is seen at the LHC.

If the SM is extended by SUSY, both these problems can be resolved as new particlesconnected to the ones of the SM by a symmetry between fermionic and bosonic degrees offreedom are introduced. This removes the quadratic divergence of the radiative correctionsof the fine tuning problem via new contributions. Additionally, if there is another newsymmetry in the SUSY model such that the lightest supersymmetric particle (LSP) is stable,the model provides a dark matter candidate. This symmetry may, for example, be thematter parity in the Minimal Supersymmetric Standard Model (MSSM) or R-symmetry inthe Minimal R-symmetric Supersymmetric Standard Model (MRSSM). In the following, themathematical basics for supersymmetry and R-symmetry are presented.

2.2. GENERAL CONSIDERATIONS ON SUPERSYMMETRY

Special relativity is imposed on a quantum field theory by means of the space-time sym-metry given by the Poincaré group. The Poincaré algebra is defined by the followingcommutation relations of its generators Pµ (translation group) and Jµν (Lorentz group) withJµν = −Jνµ:

[Pµ, Pν ] = 0 ,

[Pµ, Jρσ] = i(gµρPσ − gµσPρ) , (2.1)

[Jµν , Jρσ] = i(gνρJµσ + gµσJνρ − gµρJνσ − gνσJµρ) .

For a realistic quantum field theory, this algebra can not be enlarged non-trivially by im-posing further bosonic generators with commutation relations for a realistic quantum fieldtheory [18, 19]. Following the Haag–Łopuszanski–Sohnius theorem [20], it is however pos-sible to introduce fermionic generators with anti-commutation relations. These generatorsQiα with i = 1 . . . N, α = 1, 2 transform in the spinor representation of the Lorentz group.

4

2.2. General considerations on supersymmetry

Only N=1-supersymmetric quantum field theories can be used to describe chiral fields [21],which are needed to incorporate the Standard Model in the framework of supersymmetry.Therefore, this work is limited to N=1 SUSY models.

2.2.1. SUPER–POINCARÉ ALGEBRA

The additional anti-commutation and commutation relations for the spinor generatorsQα, Qα that closes with the Poincaré algebra (2.1) to the super–Poincaré algebra are

Qα, Qβ = 2(σµ)αβ

Pµ , Qα, Qβ = Qα, Qβ = 0 ,[Qα, Pµ

]=[Pµ, Qα

]= 0 ,

[Jµν , Qα

]= −

12 α

(σµνQ

). (2.2)

As usual, the generator Pµ gives rise to translations in space-time. It is actually possible toextend the space-time by fermionic coordinates on which Qα, Qα generate translations asmotivated by the last commutator in the super–Poincaré algebra . The details for this aregiven in the next section.

R-SYMMETRY

The super–Poincaré algebra given by eqs. (2.1) and (2.2) is actually not the most generalN=1 SUSY algebra that can be written down. It is possible to introduce a U(1) symmetry,called R-symmetry, whose generator is denoted as R and has non-vanishing commutationrelations with the Qα, Qα:†[

Qα, R]

= Qα ,[Qα, R

]= −Qα . (2.3)

All other commutators of R vanish. If this symmetry is included in a model superfields,introduced in the next section, have non-trivial transformation properties and certaininteractions normally allowed by supersymmetry are forbidden.

2.2.2. SUPERSPACE AND SUPERFIELDS

As stated before the additional generators Qα, Qα and their (anti-)commutation relationscan be interpreted as a way to extend the space-time, represented by xµ, by two spinorspaces with the Grassmann-valued elements (θα, θα) , α, α ∈ 1, 2. The resulting spaceis called superspace with elements z = (xµ, θα, θα). A SUSY transformation by ε, ε affectssuperspace coordinates as

(xµ, θ, θ)→ (xµ + iεσµθ + iεσµθ, θ + ε, θ + ε) . (2.4)

Using that θα, θα are Grassmann numbers, a field on the superspace F (z), called superfield,may be expanded into a finite series as

F (z) ≡ F (x, θ, θ) = f (x) +√

2θξ(x) +√

2 θχ(x) + θθM(x) + θθN(x)

+ θσµθAµ(x) + θθ θλ(x) + θθ θζ(x) +12θθ θθD(x) ,

(2.5)

†For a general N=M SUSY this symmetry is actually a global SU(M) symmetry, as is described in the originalpaper [20].

5

2. Supersymmetry

where f , ξ, M, N, Aµ, λ, ζ, χ and D are fields on Minkowski space. The supersymmetrytransformation of the superfield is

F → F + i(εQ + Qε)F (2.6)

and leads to the representation of the generators Qα, Qα as

Qα = i(

∂α + i σµααθα∂µ)

, Qα = −i(

∂α + iθασµαα∂µ)

. (2.7)

The representation of Pµ on Minkowski space is given as usual by the space-time derivativePµ = i∂µ. As the Qα, Qα do not commute with the derivatives ∂β and ∂β , the latter can notbe used to constrain the general superfield F in a SUSY-covariant way. SUSY-covariantderivatives may be defined as

Dα := ∂α − iσµααθα∂µ , Dα := −∂α + iθασµαα∂µ . (2.8)

The definition of R-symmetry by the commutators (2.3) on superspace follows refs. [22,23]. A generic superfield F with the arbitrary charge Qr under the U(1)R, called R-charge,transforms as

eiτRF (xµ, θ, θ)e−iτR = eiτQr F (xµ, e−iτθ, eiτθ) , (2.9)

where τ is the phase of the transformation. It can be seen that the fermionic coordi-nates θ and θ transform with an R-charge of one and minus one, respectively. With thetransformation (2.9) it is possible to find a representation of R on the superspace as

R = θα∂α − θα∂α + R . (2.10)

The operator R gives the R-charge of the component fields of F and commutes with alloperators of the super–Poincaré algebra. It can be shown easily that the representation ofR satisfies the commutation relations (2.3) with the representations for Qα, Qα.

The relevant superfields for building realistic supersymmetric models are introduced inthe following: the chiral/anti-chiral superfields and the vector superfields. For these casesalso the properties with regard to R-symmetry are discussed.

CHIRAL SUPERFIELDS

With the supersymmetry-covariant derivatives (2.8) chiral and anti-chiral superfields Φ andΦ† are defined by

DαΦ = 0 , DαΦ† = 0 . (2.11)

When using the chiral superspace coordinate yµ = xµ−iθσµθ and switching to the coordinatesystem (yµ, θα, θα), the derivative Dα becomes −∂α and a chiral superfield can be writtensimply as

Φ(y, θ, θ) = φ(y ) +√

2θψ(y ) + θθF (y ) . (2.12)

Expanded to the normal superspace coordinates, the (anti-)chiral superfields take the form

Φ(z) = φ(x) +√

2θψ(x) + θθF (x)

6

2.2. General considerations on supersymmetry

− iθσµθ∂µφ(x) −i√2θθθσ

µ∂µψ(x) +

14θθθθ∂µ∂µφ(x) , (2.13a)

Φ†(z) = φ*(x) +√

2 θψ(x) + θθF*(x)

+ iθσµθ∂µφ*(x) −

i√2θθθσµ∂µψ(x) +

14θθθθ∂µ∂µφ*(x) . (2.13b)

The Weyl-spinor ψ(x) and complex scalar field φ(x) are a pair of superpartners. The scalarfield F (x) is a complex auxiliary field. It is needed to provide additional bosonic fermionicdegrees of freedom (d.o.f.) if the on-shell equations of motion can not be used to reduce thenumber of fermionic d.o.f.. This ensures the validity of the supersymmetry transformations.

For supersymmetric extensions of the SM, matter and Higgs fields are represented bychiral superfields. The fermionic superpartner of the Higgs field is called higgsino, while thescalar superpartners of the quarks and leptons are called squarks and sleptons, respectively.

The R-charge of a chiral superfield can be chosen freely. The consequential charges ofthe component fields follow from the fact that θ has an R-charge of one. Then, the scalarcomponent field has the same R-charge as its superfield, while the fermionic componentsR-charge is reduced by one and the auxiliary fields R-charge by two.

VECTOR SUPERFIELDS

A vector superfield V (z) is defined to be a real superfield, V † = V . A general superfield (2.5)is therefore a vector superfield if f , Aµ and D are real functions and χ = ξ, N = M*, ζ = λ.

Moreover, a vector superfield may be constructed as i(Λ − Λ†), where Λ is a chiralsuperfield. Using this, a supergauge transformation of a vector superfield may be definedas

V → V + iΛ − iΛ† . (2.14)

With a certain choice of this transformation, called the Wess–Zumino gauge, it is possibleto set f , M and ξ to zero, but still to retain the freedom of an usual gauge transformationof the vector field Aµ(x). In the Wess–Zumino gauge, the vector superfield V then takesthe form

V (z) = θσµθAµ(x) + θθ θλ(x) + θθ θλ(x) +12θθ θθD(x) . (2.15)

Here, λ(x) and Aµ(x) are the set of superpartners, while D(x) is the corresponding auxiliaryfield. For supersymmetric extensions of the SM, the gauge fields are contained each in avector superfield. The field Aµ(x) is a gauge boson, while the fermionic superpartner λ(x) istypically called a gaugino.

The condition of being a real field fixes the R-charge of every vector superfield to bezero, else the transformation (2.9) could introduce a complex phase. Then, the vectorcomponent field has an R-charge of zero, as is suitable for a gauge field, and the fermionicsuperpartner has an R-charge of one.

2.2.3. SUPERSYMMETRIC LAGRANGIAN

Superfields which can contain the usual matter fields (chiral superfields) and gauge fields(vector superfields) have been defined in the previous part. In this section a description is

7

2. Supersymmetry

θα θα ∂α Dα d2θ W W aα Φ (R-charge QR) V (R-charge 0)φ ψα F Aµ λα D

1 −1 −1 −1 −2 2 1 QR QR − 1 QR − 2 0 1 0

Table 2.1. R-charges of different superspace objects appearing in the text. The last twoare the chiral and vector superfield with their respective component fields.

given how these fields can be brought together to describe interactions. A summary ofthe R-charges of elements of superspace expressions, which are needed in the following,appearing in a supersymmetric Lagrangian density is given in tab. 2.1.

The integral of a general superfield F over the full superspace yields a SUSY-invariantquantity, the action S,

S =∫

d4x∫

d2θ

∫d2θ F (x, θ, θ) , δεS = 0 , (2.16)

as the transformations by the Qα, Qα from eq. (2.7) give total derivatives, which vanishunder the integral. The integrals over the spinor spaces are denoted as

d2θ := −14

dθαdθα , d2θ := −14

dθαdθα

, d4θ := d2θ d2θ . (2.17)

For chiral superfields the SUSY-invariant action is given as

Schiral =∫

d4x∫

d2θ Φ(x, θ, θ)|θ=0 =∫

d4x∫

d4θ δ2(θ)Φ(x, θ, θ) . (2.18)

As the sum and product of superfields are also superfields, it allows to find a Lagrangiandensity as the integral over Grassmann variables

L(x) =∫

d2θ

∫d2θ F (x, θ, θ) . (2.19)

If the theory is supposed to be R-symmetric, the Lagrangian density must be invariantunder R-symmetry transformation and, as d2θ d2θ does not pick up a phase, the superfieldF must not carry any R-charge. This constrains the terms that may appear in L.

The sum and the product of chiral superfields is again a chiral superfield, respectively.Therefore, it is possible to define interactions between chiral superfields Φi for the action(2.18) via the superpotential W(Φi )

W(Φi ) = aiΦi +12µi jΦiΦj +

13!

fi jkΦiΦjΦk (2.20)

with the symmetrized coefficient matrices ai , µi j and fi jk . The superpotential contains nohigher powers in Φi as those would give rise to non-renormalizable interaction terms in theLagrangian. For a real action S, F (z) in eq. (2.16) must be a vector superfield. Thereforethe action must contain the same number of chiral and anti-chiral superfields:

SW =∫

d4x∫

d4θ[δ2(θ)W

(Φ(x, θ, θ)

)+ δ2(θ)W

(Φ†(x, θ, θ

)]. (2.21)

8

2.2. General considerations on supersymmetry

Furthermore, if the Lagrangian density is R-symmetric, the superpotential W(Φ(x, θ, θ)

)must carry an R-charge of two as δ2(θ) transforms with an R-charge of minus two. An R-sym-metric supersymmetric theory with a non-trivial superpotential must therefore contain atleast one chiral superfield that has a non-vanishing R-charge.

For gauge interactions, a non-abelian supergauge transformation for gauge multiplets ofchiral superfields is defined as

Φ→ e−iΛ(z)Φ , DαΛa(z)T a = 0 ,

Φ† → Φ†e+iΛ†(z) , DαΛa†(z)T a = 0 , (2.22)

where Λ(z) ≡ 2gΛa(z)T a, T a is the generator of the gauge group, g the respective gaugecoupling, and Λa(z) is a chiral superfield in the adjoint representation. For the vectorsuperfield V ≡ 2gV aT a, interaction terms with the chiral superfields of the form (Φ†eVΦ)may be achieved with the following gauge transformation

eV → e−iΛ†eV eiΛ , e−V → e−iΛe−V eiΛ†

. (2.23)

The term (Φ†eVΦ) by itself is already R-symmetric.

Kinetic terms of the vector superfields are defined using chiral and anti-chiral fieldstrengths Wα, W α

Wα = −14

DDe−V DαeV , Wα = 2gW aαT a , (2.24a)

W α = −14

DDeV Dαe−V , Wα

= 2gW αaT a . (2.24b)

The field strengths transform gauge covariantly and the terms Tr[WαWα] and Tr[W αW α]are gauge invariant. The integral

∫d2θ Tr

[WαWα

]respects R-symmetry as does its

conjugate.

If the superpotential W defined by eq. (2.20) is supergauge invariant under the transfor-mation (2.22), a supersymmetric Lagrangian density is given as

L =1

16g2

∫d2θ

[W aαW a

α

]|θ=0 + h. c.

+∫

d4θ Φ†eVΦ

+∫

d2θ W(Φ)|θ=0 + h. c. . (2.25)

The auxiliary F and D fields of the chiral and vector superfields may be eliminated from theLagrangian density using their respective equations of motions as

Fi = −∂W†

∂Φ†i

∣∣∣∣∣θ=θ=0

, Da = −g∑φ

φ*i T a

ijφj . (2.26)

Their contributions then take the form

LF,D = −F*i Fi −

12

∑t

Dat Da

t , (2.27)

9

2. Supersymmetry

with t running over all gauge groups. These terms give the scalar potential of a supersym-metric theory.

From all this follows that a supersymmetric Lagrangian density is defined by its gaugesymmetries, matter fields represented by chiral superfields and their superpotential.

2.2.4. SUSY BREAKING

The super–Poincaré algebra (2.2) implies that the mass operator, PµPµ, commutes withthe SUSY generators Qα, Qα[

PµPµ, Qα

]=[PµPµ, Qα

]= 0 . (2.28)

Hence, the superpartners contained in a superfield have the same mass in a fully supersym-metric theory. As this is clearly falsified by experiment, supersymmetry must be broken.To keep the positive effects of introducing supersymmetry in a theory, it is commonlyassumed that SUSY is only broken softly. Hence, only new parameters with a positive massdimension appear and no quadratic divergences in quantum corrections are introduced.

Explicitly, a softly SUSY breaking Lagrangian density is written as [24]

Lsoft,I = −φ*i (m2)i jφj +

(13!

(Af )i jkφiφjφk −12

(Bµ)i jφiφj + (Ca)iφi −12

Maλaλa + h. c.

).

(2.29)Here, φ is representing the scalar component field of a chiral superfield and λa the fermioniccomponent of a vector superfield. The a, f and µ parameter from the superpotential (2.20)appear to show the similarity of the terms. This is usually taken as the general form of aSUSY breaking Lagrangian density.

Actually, the Lagrangian density (2.27) does not have the most general form but additionalterms may be introduced [25]:

Lsoft,II =12

Dijkφ

*i φjφk − mi

aψiλa −

12

Ei jψiψj + h. c. , (2.30)

where ψ represents the fermionic component field of a chiral superfield. Terms proportionalto Di

jk are in general not soft as they can contribute with quadratic divergences to thetadpole of a complete scalar gauge singlet appearing in a model. Therefore, such a termis only introduced if this issue can be avoided. The application of R-symmetry on thesoft breaking Lagrangian densities is straightforward as no Grassmann coordinates appear.Hence, each term of eqs. (2.29) and (2.30) must have an R-charge of zero by itself. Becauseof this and the fact that the superpotential has an R-charge of two, terms appearing in thesuperpotential must not appear in a similar form in the soft breaking action as is usuallydone for supersymmetric models.

For a realistic R-symmetric theory, the Majorana mass terms for the gauginos in (2.29)have a non-vanishing R-charge and are therefore forbidden. As no massless superpartnersto the gluon or any other gaugino with a mass at the electroweak breaking scale have beenfound, there must be another source for gaugino masses. Then, Dirac masses as in thesecond term of eq. (2.30) need to be introduced.

Usually, it is assumed that soft SUSY breaking appears spontaneously such that theground state of the full SUSY theory is not supersymmetric, but SUSY is a symmetry of thefull theory. As the scalar potential in a SUSY theory is given by the F and D terms (2.27),

10

2.2. General considerations on supersymmetry

if one of them has a vacuum expectation value (vev) in the ground state it follows thatsupersymmetry is spontaneously broken. Another possibility is that the universe is in ametastable SUSY breaking state [26, 27], which is not discussed here.

If a D term acquires a vev in a SUSY theory this is called Fayet-Iliopoulos breaking [28],while for an F term it is known as O’Raifeartaigh breaking [29]. Both variants are describedin ref. [30] in more detail. There, the Nelson–Seiberg theorem [31] is noted in connectionwith O’Raifeartaigh breaking. It states that if the global minimum of a theory is broken byan F term and the superpotential is generic, then an unbroken R-symmetry exists in thetheory. As R-symmetric supersymmetry is the focus of this work, it might therefore beassumed that O’Raifeartaigh breaking would be the preferred variant for model building.As no (Majorana) gaugino mass terms, which have an R-charge of two, can arise in anR-symmetric model, the only solution is the introduction of additional fields and Dirac massterms for gauginos. Until very recently [32], this was troublesome to achieve with F termbreaking, but from ref. [33] it is known that Dirac masses for gauginos can arise if a D termfrom a U(1) symmetry acquires a vev.

This can be described by a spurion field strength Wα = θαD getting a vev 〈D〉. If it iscombined with a field strength of a gauge group W a

α, which in chiral coordinate basis up toorder θ is given as

W aα = λa

α + θαDa − α(σµσνθ)[Dµ, Dν

] a + (θθ) -terms , (2.31)

and a chiral superfieldΦ in the adjoint representation of this gauge group, given by eq. (2.12),the resulting effective terms (using the chiral coordinate basis) yields a Dirac mass for thefermionic component fields of the superfields. Additionally, the scalar component field andthe D field of the vector superfield are coupled:∫

d2θ

√2Wα

MW aαΦ

a vev=〈D〉M

∫d2θ 2(θλa)(θψa) +

√2θθDaφa − θσµσνθ︸ ︷︷ ︸

∝gµν

[Dµ, Dν

] aφa

= −MD(λaψa −

√2Daφa

), (2.32)

where M is the mass scale related to the symmetry breaking and the Dirac mass parameterMD = 〈D〉

M is introduced. The term with the commutator vanishes. Also the conjugateexpressions have to be added for the Lagrangian density and the part containing the Dterm of a vector superfield is given as follows

LD =√

2MDDa(φa + φ*a) + g∑

k

π*i

kT aij D

aπkj +

12

DaDa , (2.33)

where πk represents all scalar fields that transform non-trivially under the gauge group andit is assumed that MD is real. When solving for the equation of motion for D, the result is

Da = −

(√

2MD(φa + φ*a) + g∑

k

π*i

kT aijπ

kj

)(2.34)

and the scalar potential has the same form as eq. (2.27) when written in terms of F andD. The scalar potential receives an additional contribution from the Dirac mass parameterwhen replacing the D term by eq. (2.34). This automatically gives rise to a mass term forthe scalar component of φa but not the pseudo-scalar one. Moreover, now trilinear scalar

11

2. Supersymmetry

vertices appear with no counterpart in the MSSM but they have the structure as the Dijk

term in the Lagrangian density (2.30). In ref. [33] it was noted that the contribution (2.32) is“super-soft” such that contributions through quantum effects to other soft breaking termsare finite and do not even provide a logarithmic sensitivity on the UV scale as the softbreaking Majorana mass terms do.

Considering the other SUSY breaking terms in eq. (2.29) it is possible to generate severalof them as super-soft operators using this Dirac mass term radiatively, see ref. [33], orusing an F term vev from a gauge or gravity mediated hidden sector as done in ref. [34].As the focus of this work is of phenomenological orientation and not on the origin ofsupersymmetry breaking, only a short summary of the latter option shall be given. Thespurion of the F term that acquires a vev is denoted as X = θθF . Soft scalar masses canbe introduced via the operators∫

d4θX †XM2 Φ

†i Φi 3 m2

i |φi |2 , (2.35)

whereΦi represents the chiral superfields of the theory. Couplings of bilinears in superfieldsΦiΦj to the spinor X can yield holomorphic B mass terms in the Lagrangian (2.29) as∫

d4θX †XM2 ΦiΦj 3 Bi jφiφj , (2.36)

if allowed by all symmetries. An example for this is the Bµ term in the models discussedbelow where ΦiΦj = HuHd and also works for chiral superfields in the adjoint representation,if ΦiΦj = Tr [T T ], where T is an SU(2)L triplet. Such a term could also be generated from Dterm breaking illustrated as before by the spurion field strength Wα = θαD as∫

d2θWαWα

M2 ΦiΦj 3 B′i jφiφj . (2.37)

Mass parameters in the superpotential like the µ parameter in the MSSM can arise similarly∫d4θ

(X †

MΦiΦj

)3∫

d2θ µi jΦiΦj . (2.38)

If R-symmetry is required, it has to be ensured that the generated operator in the superpo-tential has the correct R-charge of two.

With the introduction of soft SUSY breaking terms all elements are at hand to con-struct supersymmetric models, which contain the Standard Model and can make realisticpredictions for experiments.

2.3. R-SYMMETRY FOR MODEL BUILDING

A supersymmetric model is fully specified if all its gauge symmetries, the chiral superfieldcontent and respective representation under the gauge symmetries, the correspondingsuperpotential and the soft SUSY breaking Lagrangian density are given. Further, the use ofglobal symmetries, which constrain the superpotential or soft breaking terms, is possibleor even required for model building. The introduction of global symmetries is necessary forexample to avoid terms in the Lagrangian density leading to rapid proton decay.

12

2.3. R-symmetry for model building

If supersymmetry is applied to the SM, the SU(3)C × SU(2)L × U(1)Y gauge symmetryis taken over, introducing a vector superfield for each gauge field. As all matter fields ofthe SM transform in a trivial or fundamental representation they can not be contained inthese vector superfields. Rather, for each matter field a chiral superfield is required. Specialcare needs to be taken with the Higgs field, as with only one single chiral Higgs superfieldthe fermionic component field would give rise to gauge anomalies. Therefore, a secondHiggs doublet superfield with the opposite hypercharge is introduced. This also allowsto include terms in the superpotential that, after electroweak symmetry breaking, yieldmasses for all the fermionic matter fields. A summary of all the superfields of this minimalsupersymmetric extension of the SM field content is given in tab. 2.2.

The general superpotential and SUSY breaking Lagrangian density of these fields arevery lengthy and in the following only those in agreement with further global symmetriesare described. It has been shown before that a supersymmetric theory may also containan R-symmetry, which for (N=1) SUSY is a global U(1) symmetry, where the charges of thecomponent fields of a certain superfield are related as described in tab. 2.1. The introductionof R-symmetry maximizes the symmetry content of the model and is a well-motivatedapproach to model building. A natural idea would be to impose this symmetry on thesuperpotential and SUSY breaking Lagrangian density of the superfield content given intab. 2.2.

With R-symmetry imposed, R-charges need to be assigned to the chiral superfields.One possibility is to assume that all the SM matter fields carry an R-charge of zero asdo both scalar Higgs fields. Alternatively, the non-SM-like Higgs field Hd might have theR-charge two to allow for a mass term of the Higgs fields in the superpotential. Then,superpotential couplings of this field to the down-type matter fields are forbidden and thecorresponding fermion masses have to be generated radiatively. The Higgs field Hd cantherefore not acquire a vev, which would break R-symmetry [35]. This corresponds to ascenario called the tanβ =∞ limit in the MSSM, which has been investigated in ref. [36].Another possibility is to identify the R-charge with the lepton number L of one flavor andone of the sneutrinos acquire a vacuum expectation value giving mass to the down-typefermions. This scenario has been studied in refs. [37–40]. Each scenario has a differentsuperpotential as the R-charge assignment forbids certain terms from appearing, if they donot have an R-charge of two. Option one has the following superpotential

WI = −Y eij

(HdLi

)E j − Y d

ij

(HdQi

)Dj + Y u

ij

(HuQi

)U j , (2.39)

while for the second option it takes the form

WII = Y uij

(HuQi

)U j + µ(HdHu) , (2.40)

and for the third option

WIII = Y uij

(HuQi

)U j − Y d

ij

(LaQi

)Dj − Y b (LaLb

)Eb − Y c (LaLc

)Ec , (2.41)

where a, b, c = e,µ, τ by convention of ref. [37].

The problem of giving masses to the gauginos which carry now a non-zero U(1) chargeand, therefore, can not be of Majorana type, remains for all these variants. There aretwo ways to circumvent this. The first one is usually considered and leads to the MinimalSupersymmetric Standard Model (MSSM). The set of unbroken symmetries of the model

13

2. Supersymmetry

Superfield Component fields SU(3)C×spin 0 spin 1/ 2 spin 1 SU(2)L × U(1)Y

Qi = (Qui Qdi )T qiL = (uiL diL)T qiL = (uiL diL)T (3, 2, 1

6 )U i u*

iR uCiR (3, 1, −2

3 )Di d*

iR dCiR (3, 1, 1

3 )Li = (Lνi Lei )

T ˜iL = (νiL eiL)T `iL = (νiL eiL)T (1, 2, −12 )

E i e*iR eC

iR (1, 1, 1)Hd = (H0

d H−d )T hd = (h0

d h−d )T hd = (h0

d h−d )T (1, 2, −1

2 )Hu = (H+

u H0u )T hu = (h+

u h0u)T hu = (h+

u h0u)T (1, 2, 1

2 )

ga ga gaµ (8, 1, 0)

W j W j W jµ (1, 3, 0)

B B Bµ (1, 1, 0)

Table 2.2. Shown are the superfields of the MSSM and their respective component fieldsas well as their group representation under SU(3)C × SU(2)L × U(1)Y . For U(1)Ythe quantum number Y/ 2 is given. The index i ∈ 1, 2, 3 runs over the threegenerations of the matter fields. The charge conjugated spinor field ψC is derivedfrom ψ via the relation ψC = C ψ

Twith C = iγ2γ0.

needs to be reduced for this approach. Then, not the full R-symmetry is included as a globalsymmetry of the model but only a discrete subgroup of it. The Z2 subgroup is commonlycalled R-parity. The MSSM is discussed in the next section.

The second option to generate gaugino masses is the inclusion of additional degrees offreedom in the model as they are required to write down Dirac masses for gauginos. ASUSY model, the Minimal R-symmetric Supersymmetric Standard Model (MRSSM), whichdoes just this is described in the last section of this chapter.

2.4. THE MINIMAL SUPERSYMMETRIC STANDARD MODEL

In the following the Minimal Supersymmetric Standard Model (MSSM) is described shortlyas it is the supersymmetric model most commonly investigated and is used for comparisonwith the R-symmetric SUSY model which is in the focus of this work.

2.4.1. SYMMETRIES AND FIELDS

As described in the last section, the MSSM has the same gauge symmetry as the SM,which is SU(3)C × SU(2)L × U(1)Y with the gauge couplings g3, g2, g1, respectively. Thefield content and the respective representations under the gauge symmetries are given intab. 2.2. R-parity is usually specified as an additional global symmetry.

R-PARITY

R-parity can be understood as the Z2 subgroup of R-symmetry, where the continuousparameter is fixed to τ = π. Therefore, all the fermionic coordinates of superspacetransform with an odd parity. The parity assignment on the superfield content is suchthat SM component fields have an even parity. From this follows immediately that their

14

2.4. The Minimal Supersymmetric Standard Model

superpartners have an odd parity. This can also be put in the form of the so-called matterparity for the component fields

PM = (−1)3(B−L)+2S . (2.42)

Here, B, L and S are the baryon-number, lepton-number and spin operator, respectively.From a phenomenological point of view, R-parity is introduced to forbid baryon and leptonnumber violating operators including the ones needed for proton decay. Another conse-quence is the existence of a lightest stable supersymmetric particle (LSP), as the parityforbids a further decay of the lightest state with an odd parity. The LSP is a dark mattercandidate of the model.

Taking into account all gauge symmetries and R-parity the following superpotential is themost general one allowed for the MSSM:

WMSSM = −Y eij

(HdLi

)E j − Y d

ij

(HdQi

)Dj + Y u

ij

(HuQi

)U j + µ(HdHu) , (2.43)

here the SU(2)L invariant product is given as(HQ)

= HαQβεαβ with ε taken from (A.2). This

fixes all supersymmetric interactions of the fields. The Y e, Y d , Y u are 3 × 3 matrices in thecorresponding flavor space.

2.4.2. SUSY BREAKING

The general soft SUSY breaking Lagrangian density of the MSSM is usually chosen to be

LMSSMsoft = − m2

qL,i j q*iLqjL − m2

uR ,i j u*iRujR − m2

dR ,i j d*iRdjR − m2

˜L,i j˜*iL

˜jL − m2eR ,i j e*

iRejR

− m2hd

|hd |2 − m2hu

|hu|2 −(Bµ(hdhu) + h. c.

)−[

(Ayd )i j (hd qiL)d*jR + (Aye)i j (hd ˜iL)e*

jR − (Ayu)i j (huqiL)u*jR + h. c.

]−

12

(M3gaga + M2W l λl + M1BB + h. c.

).

(2.44)

As in the superpotential, all parameters connected to the sfermions are taken as general3 × 3 matrices in flavor space. Following ref. [25], this is not the most general form possiblebut additional terms are allowed:

LMSSMsoft, add = −(Dyd )i j (h*

uqiL)d*jR + (Dye)i j (h*

u˜iL)e*

jR − (Dyu)i j (h*d qiL)u*

jR

+12

Mh(hd hu) + h. c. .(2.45)

One of the three parameters µ, Bµ and Mh is actually redundant and can be removedby appropriate redefinition of the other two. The additional terms of LMSSM

soft, add are usuallyneglected and, as the MSSM is not in the focus of this work, they are not explored furtherhere.

15

2. Supersymmetry

2.4.3. ELECTROWEAK SYMMETRY BREAKING

Following the layout of the SM, also in the MSSM the gauge group SU(3)C × SU(2)L × U(1)Yis spontaneously broken to SU(3)C × U(1)em when the neutral Higgs scalars acquire vevs:

h0d =

1√2

(vd + φd + iσd ) , h0u =

1√2

(vu + φu + iσu) .

The scalar potential and tadpole equations for spontaneous symmetry breaking and ensuingmass matrices of the MSSM are not given here, as they can be found in numerousreferences [30, 41]. Here, only the most relevant relations are pointed out that are neededto make the connection to the SM.

The vev of the SM Higgs field is given by v2 = v2d + v2

u , additionally the parametertanβ = vu/ vd may be defined. Then, the tree-level masses of the gauge bosons are givenas

m2W =

g22

4v2 , m2

Z =g2

1 + g22

4v2 . (2.46)

The masses of the additional CP-odd and charged Higgs boson are

m2A = Bµ

(tanβ +

1tanβ

), m2

H± = m2A + m2

W , (2.47)

respectively, while the masses of the Goldstone bosons that arise from the CP-odd andcharged Higgs boson mass matrices depend on the gauge fixing as in the SM. Both massmatrices are diagonalized by using β as mixing angle. The CP-even Higgs boson massmatrix is diagonalized by the angle α.

The MSSM is the simplest realistic supersymmetric extension of the SM. Because ofthis, its phenomenology is well explored in many avenues [30, 41]. The non-discoveryof SUSY at the LHC so far pushing the mass scale of the superpartners in the multi-TeVrange and the 125 GeV Higgs boson mass leads to the “little hierarchy problem” [42] of theMSSM. Its interpretation as a problem of the MSSM is ambiguous, but one can draw theconclusion that the MSSM is not necessarily elevated above other realistic SUSY models.One possible avenue is the inclusion of R-symmetry to study the full symmetry content ofthe super–Poincaré algebra.

2.5. R-SYMMETRIC SUPERSYMMETRY

As discussed in section 2.3, a SUSY model with R-symmetry requires an extended fieldcontent to allow for Dirac mass terms of the gauginos. Furthermore, appropriate massterms for the higgsinos need to be introduced. This may be done by choosing one ofthe Higgs doublets to have an R-charge of two, which gives rise to the superpotential(2.40). This superpotential contains the usual µ term but down-type Yukawa couplings areforbidden, if the down-type Higgs doublet is charged under R-symmetry. Hence, all massesof down-type fermions need to be generated by radiative corrections. Another option isto enlarge the field content by two R-charged Higgs doublets, which can be motivatedas an N=2 SUSY extension of the MSSM Higgs sector. The latter model is describedin the remainder of this section and is the basis for the phenomenological studies donewithin this work. It is known as the Minimal R-symmetric Supersymmetric Standard Model(MRSSM) [34]. This and several models containing R-symmetry have been proposed and

16

2.5. R-symmetric Supersymmetry

some of their aspects studied before, see refs. [43–71].

2.5.1. SYMMETRIES AND FIELDS

The MRSSM has the same gauge symmetry SU(3)C ×SU(2)L ×U(1)Y as the SM and MSSM.Moreover, it has an unbroken global U(1)R symmetry. The model contains all superfieldsof the MSSM listed in tab. 2.2. Further chiral superfields are introduced to allow for Diracgaugino and higgsino mass terms. These are adjoint chiral superfields O, T , S for eachgauge group with R-charge zero and two Higgs chiral superfields, which have the samequantum numbers as the usual Higgs superfields except for their R-charge of two. Thegauge symmetry properties of the new superfields are given in tab. 2.3 and the R-chargesof all component fields in the MRSSM are listed in tab. 2.4.

Superfield Component fields SU(3)C×spin 0 spin 1/ 2 SU(2)L × U(1)Y

Ru = (R0u R−

u )T Ru = (R0u R−

u )T Ru = (R0u R−

u )T (1, 2, −12 )

Rd = (R+d R0

d )T Rd = (R+d R0

d )T Rd = (R+d R0

d )T (1, 2, 12 )

Oa Oa Oa (8, 1, 0)

T =

(T0/√

2 T+

T− −T0/√

2

)T =

(T0/√

2 T+

T− −T0/√

2

)T =

(T0/√

2 T+

T− −T0/√

2

)(1, 3, 0)

S S S (1, 1, 0)

Table 2.3. Shown are the additionally introduced chiral superfields of the MRSSM andtheir respective component fields as well as their group representation underSU(3)C × SU(2)L × U(1)Y . For U(1)Y the quantum number Y/ 2 is given. It shouldbe noted that the two components of the triplet charged under U(1)em areindependent as T *

− 6= T+.

Superfield Boson Fermion

g, W, B 0 gµ, W µ, Bµ 0 g, W , B +1L, E +1 ˜L, e*

R +1 `L, eCR 0

Q, D, U +1 qL, d*R, u*

R +1 qL, dCR , uC

R 0Hd,u 0 hd,u 0 hd,u −1Rd,u +2 Rd,u +2 Rd,u +1O, T , S 0 O, T, S 0 O, T , S −1

Table 2.4. The R-charges of the superfields and the corresponding bosonic and fermioniccomponents of the MRSSM are collected in this table.

This choice of R-charges leads to the superpotential (2.39) for the MSSM superfields.Additional terms involving the new chiral superfields are also allowed. R-symmetry for-bids the µ term of the MSSM but allows for bilinear combinations of the normal HiggsSU(2)L doublets Hd and Hu with the R-charged Higgs SU(2)L doublets Rd and Ru in the

17

2. Supersymmetry

superpotential:

WMRSSM,µ = µd (RdHd ) + µu (RuHu) . (2.48)

These combinations yield Dirac masses for the higgsinos. Two sets of trilinear terms withthe adjoint electroweak chiral superfields T , S are also allowed:

WMRSSM,λ = λd S (RdHd ) + λu S(RuHu)

+Λd (Rd T ) Hd + Λu(RuT ) Hu .(2.49)

For completeness, the full superpotential of the MRSSM considered in this thesis is

WMRSSM =µd (RdHd ) + µu (RuHu) + Λd (Rd T ) Hd + Λu (RuT ) Hu

+ λd S (RdHd ) + λu S (RuHu) − Yd D (QHd ) − Ye E (LHd ) + Yu (U QHu) .(2.50)

R-symmetry does naturally not allow for any baryon- or lepton-number violating contribu-tions in the superpotential (as well as dimension-five operators mediating proton decay)[72, 73].

N=2 SUSY RELATIONS

When including additional chiral superfields in the adjoint representation in an N=1 SUSYgauge theory with U(1)R symmetry, it actually becomes N=2 supersymmetric. The compo-nent fields of the chiral superfield are the required degrees of freedom necessary for anN=2 gauge supermultiplet, as given in chapter 27.9 of ref. [74]. Therefore, the part of theMRSSM consisting only of the gauge superfields and the singlet, triplet and octet adjointchiral superfields is N=2 supersymmetric.

The N=2 supersymmetry may actually be extended to the Higgs and R-Higgs superfieldsof the MRSSM. Each of a set of up or down Higgs and R-Higgs doublet can be combinedinto an N=2 hypermultiplet. The µ-like contributions to the superpotential (2.48) respectthis symmetry. On the other hand, for the Yukawa-like λ and Λ contributions in eq. (2.49)relations to the gauge couplings are required by the N=2 SUSY such that

λd = −λu =g1√

2, Λd = Λu = g2 , (2.51)

as the singlet and triplet are part of the N=2 gauge multiplets.However, it is not possible to extend the N=2 SUSY to matter superfields, as in nature

the different chiralities of leptons and quarks are in separate representations of the SU(3)C ×SU(2)L × U(1)Y gauge group. The prediction of N=2 SUSY would be the existence of statesof both chiralities from a hypermultiplet in one representation with the same mass. As thisis clearly not the case, N=2 supersymmetry would need to be broken at a high scale toN=1 such that the chiral structure of the SM can be matched [56, 62]. Such a breaking ofN=2 SUSY would also lead to the Λ/λ parameters having different values than the gaugecouplings at a lower scale as the conditions (2.51) would receive threshold correctionsand the parameters would run differently. The study of the MRSSM as an N=1 SUSYmodel with R-symmetry is therefore also a phenomenological study of a realistic N=2SUSY model at the TeV scale as it accounts for the chirality of matter but also respects theN=2 symmetry as far as possible.

18

2.5. R-symmetric Supersymmetry

2.5.2. SUSY BREAKING

When considering supersymmetry breaking in the MRSSM obeying R-symmetry, it isclear that all the soft breaking masses of the MSSM scalar fields can be taken over fromeq. (2.44). The same holds true for the holomorphic bilinear term of the MSSM Higgsdoublets. The trilinear A terms of eq. (2.44) are forbidden as the sfermions have anR-charge and these terms are therefore not R-invariant. This relaxes the flavor problem ofthe MSSM [34, 43]. As discussed before, the same applies to the Majorana masses of thegauginos. This leaves MSSM-like contributions as follows

LMRSSMsoft, MSSM-like = − m2

qL,i j q*iLqjL − m2

uR ,i j u*iRujR − m2

dR ,i j d*iRdjR − m2

˜L,i j˜*iL

˜jL − m2eR ,i j e*

iRejR

− m2hd

|hd |2 − m2hu

|hu|2 −(Bµ(hdhu) + h. c.

).

(2.52)

As described in subsection 2.2.4, Dirac mass terms linking the fermionic componentsof a vector and a chiral superfield can be created via D term breaking of a hidden sectorU(1)′, see ref. [33]. This leads to additional terms connecting the scalar component field ofthe chiral superfield and the D field of the vector superfield as in eq. (2.32). In the MRSSMthis is done for all three gauge groups so that the Lagrangian density containing the Diracmasses reads

LMRSSMsoft, Dirac = − MD

B (B S +√

2DB S) − MDW (W aT a +

√2Da

W T a) − MDO(gaOa +

√2Da

gOa) + h. c. .

(2.53)

This modifies the on-shell equations of motion for the auxiliary D terms as given in eq. (2.34).

Following ref. [25], it is noteworthy that the term DBS in general lead to quadraticdivergences in the tadpole equation for the scalar singlet so that this contribution shouldactually not be categorized as soft. The quadratic divergence does not arise as thehyper-charge of each generation of matter fields and of all Higgs and R-Higgs doubletfields sums up to zero. The cancellation seems comparable to the one of the hyper-chargegauge anomaly in the SM and MSSM. It is possible that this connection may be explainedusing the N=2 SUSY relations described before, but studying this is outside the scope ofthis work.

For the adjoint scalars S, T and O and the R-Higgs bosons non-holomorphic quadraticmasses can be generated as for the MSSM fields:

LMRSSMsoft, add = −m2

Rd|Rd |2 − m2

Ru|Ru|2 − m2

S |S|2 − m2T Tr(T *T ) − m2

O Tr(O*O) . (2.54)

Additional holomorphic terms are allowed involving the adjoint scalars and Higgs bosonsas

LMRSSMsoft, holomorph =BsSS + Bt Tr(TT ) + Bo Tr(OO) + CsS

+ AshS(hdhu) + AsoS Tr(OO) + AsS3

+ Ath(hdT )hu + AstS Tr(TT ) + AtTr (TTT ) + h. c. .

(2.55)

As in the MSSM, the class of holomorphic interactions does not give all soft breaking terms

19

2. Supersymmetry

but non-holomorphic terms involving the R-Higgs bosons and sfermions are possible:

LMRSSMsoft, non-holomorph = −(Dyd )i j (R*

uqiL)d*jR +(Dye)i j (R*

u`iL)e*jR −(Dyu)i j (R*

dqiL)u*jR +h. c. . (2.56)

For simplicity, the last two categories are not considered in this thesis, so that the pa-rameter space can be investigated in a meaningful way. For the influence of some of theholomorphic mass terms see also ref. [75].

Thus the full soft breaking Lagrangian density of the MRSSM considered in this work isgiven as

LMRSSMsoft = LMRSSM

soft, MSSM-like + LMRSSMsoft, Dirac + LMRSSM

soft, add (2.57)

2.5.3. ELECTROWEAK SYMMETRY BREAKING

Since R-Higgs bosons carry an R-charge of two, they do not develop vacuum expectationvalues. For electroweak symmetry breaking (EWSB) only the electroweak scalars with noR-charge develop a vev:

h0d =

1√2

(vd + φd + iσd ) , h0u =

1√2

(vu + φu + iσu) ,

T 0 =1√2

(vT + φT + iσT ) , S =1√2

(vS + φS + iσS) .

Additionally, the color octet has to be split into its CP-even and odd part

O =1√2

(φO + iσO) , (2.58)

as they obtain different contributions from the gluino Dirac mass parameter in eq. (2.53).

MASSES IN THE MRSSM

All mass matrices of the MRSSM are given in Appendix C.2. The information not giventhere are the masses of the electroweak bosons, which are the same as in the MSSM,except that the W boson mass receives an additional contribution from the triplet vev:

m2W =

g22

4v2 + g2

2v2T , m2

Z =g2

1 + g22

4v2 . (2.59)

The mass of the Dirac gluino, which consists of the two Weyl-spinors g and O, is just theDirac mass parameter:

mg = MDO . (2.60)

The masses of the scalar and pseudo-scalar octet are

mφO =√

m2O + 4(MD

O)2 , mσO = mO . (2.61)

2.5.4. SYMMETRY TRANSFORMATIONS OF THE MRSSM LAGRANGIAN

Fields of a field theory are not physical quantities and may be redefined. This redefinitionallows to absorb complex phases of parameters in the Lagrangian density. This retains onlyphysical phases accessible by experiment. A well known example is the CKM matrix in

20

2.6. Summary

the SM [76]. Similarly, it is possible to change the phases of fields in the MRSSM, which isexplored in the following.

The usual Peccei–Quinn transformation of

hd/ u → exp(iα)hd/ u ,

X → exp(−iα)X ; X ∈ Rd/ u, UQ, DQ, EL (2.62)

leads to a phase transformation of Bµ leaving the rest of the Lagrangian invariant and thusallows to choose Bµ real and positive.

Transformations of the R-Higgs bosons can be used to eliminate a phase among each ofthe sets: µu,Λu,λu and µd ,Λd ,λd , making it possible to choose the µ parameters realand positive. These are:

Rd/ u → exp(iα)Rd/ u ,

Ω→ exp(−iα)Ω; Ω ∈ µd/ u,Λd/ u,λd/ u . (2.63)

Transformations of the singlet and triplet can be used to eliminate a phase among eachof the sets: MD

B ,λu,λd and MDW ,Λu,Λd , making it possible to choose the Dirac mass

parameters as real and positive. The transformations read

S → exp(iα)S ,

Ω→ exp(−iα)Ω; Ω ∈ MDB ,λu,λd (2.64)

and

T → exp(iα)T ,

Ω→ exp(−iα)Ω; Ω ∈ MDW ,Λu,Λd . (2.65)

This carries consequences for the phase of the vevs coming from the terms with D fields.The MRSSM tadpole equations are given in the appendix in eq. (C.3). For simplicity, all

parameters are set real and only relative signs regarding the vevs are explored. Looking atthe first two tadpole equations for vd and vu, changing the sign of Bµ can be compensatedby changing the sign of one of the vevs, thus changing the sign of tanβ. This is comparableto the MSSM and is reflected by the CP-odd Higgs mass formula. As with the Lagrangian,changing the sign of µu or µd in the tadpole equations can be compensated by changingthe sign of the associated λ and Λ.

A sign change of the Dirac masses can be absorbed by changing the sign of vS (for MDB ) or

vT (for MDW ) and the associated λ and Λ, keeping the tadpole equations invariant. Therefore,

to fix the gaugino masses to positive values, negative vS and vT must be allowed. All othersoft masses need to be positive to avoid unwanted charge or color symmetry breaking.

2.6. SUMMARY

The MSSM is a well established supersymmetric extension of the SM and has been studiedin great detail in the last decades. In this model the neutralinos and gluino are Majoranafermions. This means that these states are their own antiparticles, which leads to specificpredictions of the MSSM.

21

2. Supersymmetry

One of the prominent features of an R-symmetric model like the MRSSM is the prohibi-tion of Majorana mass terms for such fields. As massless neutralinos and gluinos are inconflict with experimental data Dirac mass terms need to be introduced. This requires theinclusion of additional degrees of freedom. Then, the neutralinos and gluino are of Diractype and particle and corresponding antiparticle can be distinguished by a difference inR-charge. Moreover, as SUSY requires the same number of fermionic and bosonic degreesof freedom additional bosons are introduced. This leads to an extended Higgs sector in theMRSSM. Because of the Dirac nature of neutralinos and gluino and the additional scalarstates the MRSSM predicts a well-motivated phenomenology distinct from the MSSM.

Moreover, even as the MSSM solves many of the SM’s problems, it is not without itsown set of issues. These can be solved with the introduction of R-symmetry.

The soft SUSY breaking trilinear couplings appearing in eq. (2.44) may lead to largecontributions to flavor violating observables if the parameters have an arbitrary structure.R-symmetry does not allow these terms and it has been shown that the Dirac nature of theneutralinos and gluino may bring additional benefits for flavor observables [34, 43] (But seeref. [77] for caveats beyond the mass insertion limit). Thus, the predictions of the MRSSMare generically in agreement with flavor data even when considering scenarios with ananarchic flavor pattern and for sfermion masses below the TeV scale.

The MSSM allows for baryogenesis with a first order phase transition only in constrainedparameter regions when experimental limits [78] are included. In an R-symmetric modelthe required phase transition can also achieved for more open parameter regions [44].Similarly, heavy Dirac gluinos suppress the production cross section for squarks, makingsquarks below the TeV scale generically compatible with LHC data [45].

In this chapter, the MRSSM as a viable SUSY model has been introduced at the classicallevel without considering further quantum effects. For the phenomenological study of itsphysics in the latter part of this thesis quantum field theoretical methods are required. Theyare introduced in the following chapter and the computational implementation is discussed.

22

3. INPUT PARAMETERS FOR THEMRSSM

A supersymmetric model at the TeV scale as a possible description nature needs to beinterpreted in the context of quantum field theory. The MRSSM as a viable supersymmetricmodel has been introduced and motivated in the last chapter. For the discussion of itsphenomenology in the following chapters, it is crucial to derive precise predictions forexperiments with small theory uncertainties. Experimental results are usually interpretedin the context of the SM. This experimental data is required as input to derive predictionsfrom the MRSSM. Hence, to achieve a meaningful connection to experiment, a consistentmatching to the SM is needed.

The first part of this chapter contains a description of the quantum field theoreticalfoundation. Then, the renormalization of the MRSSM and its matching to the SM isdiscussed, which fixes the required input parameters. From there, it is possible to calculatethe mass spectrum of the model, which is the basis for all following predictions. In thelast part the technical implementation is presented and phenomenologically interestingscenarios are specified for further study within this work. The quantum field theoreticalfoundation comprehensively presented here is based on refs. [79–82].

3.1. FIELD THEORETICAL CONCEPTS

The study of an extended model describing nature, like the MRSSM, requires the applicationof a multitude of quantum field theoretical concepts to relate the model parameters tophysical quantities and draw meaningful conclusions to make predictions for experiments.The following sections provide an overview of the most relevant methods used to connectthe MRSSM to experiment.

3.1.1. REGULARIZATION AND RENORMALIZATION

Renormalization is needed when calculating observables in a quantum field theory. Usingthe perturbation series of scattering amplitudes with small gauge couplings gives finiteresults at the lowest order. The inclusion of higher-orders corrections by adding Feynmandiagrams with loops introduces the integration over internal momenta to capture all quan-tum field fluctuations. Doing this in a naive way leads to divergent results. Divergences

23

3. Input parameters for the MRSSM

originating from the high momentum regime are called ultraviolet (UV) divergences, if theyappear in the limit of zero momentum they are called infrared (IR) divergences.

It could be assumed that such a non-convergent result has no predictive power. It is,however, possible to treat these divergences in a formal way using regularization. Thishandles the infinities consistently and finite predictions for observables can be calculatedat the end. Then, the fact that parameters of a Lagrangian density do not have to be finitequantities but rather can be renormalized to absorb the regularized UV divergences has tobe used. IR divergences cancel each other if an observable is correctly defined as statedby the Kinoshita–Lee–Nauenberg theorem [83, 84].

A theory is called renormalizable if all UV divergences can be absorbed in all ordersof perturbation theory by the finite set of parameters in the Lagrangian density, and itis assumed that the Lagrangian density of the MRSSM satisfies this condition. In thefollowing, the regularization and renormalization schemes used in this work are introducedand their application for the MRSSM is presented.

REGULARIZATION

Many different ways exist to regularize a divergent loop integral in a quantum field theorycalculation. All of them have the introduction of an unphysical dimensionful parameter incommon. The result of a calculation to all orders in perturbation theory must not dependon this parameter for the renormalization to work. The following list discusses somewell-known regularization schemes:

Cut-off regularization The most straight-forward regularization approach is the introduc-tion of a cut-off scale Λ so that the magnitude of integration momentum k does notfluctuate to infinity. This can be achieved by the replacement of the momentumintegral as ∫ ∞

0d k →

∫ Λ0

d k . (3.1)

Divergences are regularized as polynomial in Λ or logΛ with this replacement andintegrals give UV finite results. The drawback of this method is the violation of sym-metries of the underlying theory, most obviously the violation of translation invarianceof k → k + p under the integral, but also the violation of gauge symmetry. Neverthe-less, it allows to count the superficial degree of divergence of loop contributions asthe highest exponent of Λ.

Dimensional regularization [85, 86] A regularization scheme preserving Lorentz and gaugeinvariance is the dimensional regularization scheme, where the four dimensional mo-menta are expanded to D dimensions and the integral becomes∫

d4 k(2π)4

→ Q4−D∫

dD k(2π)D

. (3.2)

The dimensionful parameter Q is introduced to keep the dimensionality of the wholeexpression fixed. Formal definitions of dimensional regularization and further applica-tions may be found in ref. [87]. Divergences are regularized in inverse powers of theparameter ε = (4 − D)/ 2. The preservation of gauge symmetry is the reason why thismethod is commonly used in the renormalization of gauge theories. Using relationsonly valid in four dimensions when applying dimensional regularization may lead to

24

3.1. Field theoretical concepts

inconsistencies as is for example known with the γ5 problem, see ref. [88] for anoverview.

Dimensional reduction [89–94] Another drawback of dimensional regularization is thebreaking of supersymmetry as it leads to a mismatch of the fermionic and bosonicdegrees of freedom when vector fields are D-dimensional. The solution (at least upto the two-loop order for certain processes [95]) is the introduction of dimensionalreduction. Here, only the momenta are extended to D dimension, while all otherquantities stay four-dimensional. For formal calculation there is still the need to extendvector bosons to D dimensions but this is achieved by separating the four-dimensionalfield into a D-dimensional part and a 2ε-dimensional one, corresponding to so-calledε scalars. In non-supersymmetric theories the couplings of the ε scalars and theircorresponding renormalization need to be treated independently from the gaugebosons while, if SUSY is present, it enforces relations to the gauge couplings. Seeref. [96, 97] for further subtleties how to deal with dimensional regularization andreduction.

From the previous points it becomes clear that dimensional reduction is the regularizationof choice for a calculation in a supersymmetric theory. The SM is not supersymmetric andit is usually easier to employ a dimensional regularization scheme. Hence, all the modelparameters extracted from experimental data are usually given as results of a calculation indimensional regularization, see e.g. ref. [98]. Therefore, when calculating parameters indimensional reduction the translation between the two schemes is necessary.

RENORMALIZATION

Parameters appearing in a Lagrangian density should not be seen as physical observables,but are connected to them by the calculation of Green’s functions and S-matrix elements.Moreover, there is a freedom to rescale them as

λ→ λ0 = Zλλ = λ + δλ , δλ = (Zλ − 1)λ , (3.3)

where λ stands for a generic parameter in the model, the subscript 0 denotes that λ0 is anunrenormalized, so-called bare, parameter. The parameter δλ is called a counterterm.

Similarly, this can be done for the fields of the theory, called wave function renormaliza-tion:

ψ → ψ0 =√

Zψψ =√

1 + δZψψ , (3.4)

where analogously ψ is a generic field of the model and δZψ the corresponding counterterm.In a renormalizable theory all appearing UV divergences can be absorbed into the Z

factors of the finite set of parameters and fields to all loop orders. In the Lagrangiandensity, the parameters and fields of the quantum field theory can be expanded into twocomponents. One is the renormalized part which has the same form as the unrenormalizedLagrangian density but with the bare fields replaced by renormalized ones. The other partis the so-called counterterm Lagrangian density which consists of all terms containingcounterterms. With the expanded Lagrangian density the usual Feynman diagrammaticcalculations can be carried out. The counterterms then appear as additional vertices and aredifferent from zero if higher-order corrections are considered. The elements of this seriesare needed to absorb the UV divergences, which have been regularized, order-by-order.

25

3. Input parameters for the MRSSM

This fixes the UV divergent structure of the counterterms but still leaves the freedom tochoose the finite part. Two possibilities are usually considered for this:

Modified minimal subtraction The most straight forward choice for the finite part is toset it to zero, so that the renormalization constants are purely UV divergent. Thisis known as minimal subtraction. Starting at the one-loop level, renormalizationconstants are therefore proportional to the pole in epsilon as

δλ1L,MS ∝ 1ε

∣∣∣∣dim. reg.

, δλ1L,DR ∝ 1ε

∣∣∣∣dim. red.

. (3.5)

The superscript MS stands for minimal subtraction when dimensional regularizationis used. In case of dimensional reduction as regularization the superscript DR is usedwhen minimal subtraction is the renormalization scheme.

All regularized loop integrals contain common terms that accompany the regulator.This fact is used to define modified minimal subtraction schemes. Then, the finitepart is set to this common expression as

δλ1L,MS ∝ 1ε

− γE + log 4π∣∣∣∣dim. reg.

, δλ1L,DR ∝ 1ε

− γE + log 4π∣∣∣∣dim. red.

, (3.6)

where γE is the Euler–Mascheroni constant. This renormalization can also be phraseddifferently. With the knowledge that the theory is renormalizable it is not necessaryto calculate the counterterms explicitly. It is sufficient to set the common factor1ε − γE + log 4π to zero in all results from loop integrals appearing in the calculationdoing an implicit renormalization.

On-shell renormalization When a parameter is renormalized on-shell all higher-ordercontributions are absorbed into its counterterm. Then, the tree-level value givesthe full result (up to the loop-order accuracy actually used in the calculation) andit is possible to find a direct relation between the renormalized parameters andobservables. It is common to renormalize mass parameters and the wave functionson-shell as it simplifies the transition from Green’s functions to S-matrix elementsusing the LSZ reduction formula [99]. Then, the mass parameter can be identifiedwith the pole mass of the field and the residuum of the propagator is one.

To fix the notation for the remainder of the thesis, only three types of parameters arerenormalized on-shell: Masses mi , coupling constants αj and the electroweak mixingangle θ. These on-shell renormalized parameters are always given without carets whereastheir DR or MS renormalized counterparts are denoted with a caret. As an example, m2

Zdenotes the Z boson pole mass while m2

Z is the DR renormalized mass parameter. Theseparameters are connected as

m2Z = m2

Z − ΠT(

p2 = m2Z

), (3.7)

where ΠT is the transverse part of theDR renormalized self-energy of the Z boson withthe momentum taken at the on-shell mass. The DR scheme is used as the standardrenormalization choice for all other parameters. If an ambiguity arises, when for examplegiving transformation rules, the corresponding scheme is given additionally by superscript.

26

3.1. Field theoretical concepts

3.1.2. EFFECTIVE ACTION

Green’s functions are used as a representation of particles and their interactions in aquantum field theory and can be calculated using the path-integral method. For this, thegenerating functional Z of all Green’s functions is defined as

Z[J ] =∫

Dφexp i(∫

d x4L(φ(x), φ(x)) + J (x)φ(x))

, (3.8)

where φ(x) symbolizes all fields in the interaction picture and L the Lagrangian density ofthe theory. The classical sources J (x) are introduced for each species of field so that theterm J φ represents the sum over the product of all fields with their corresponding source.The measure Dφ sums over all possible configurations of the fields. Varying Z[J ] withrespect to n sources gives the sum of all Feynman graphs with n external legs:

δnZ[J ]δ(iJ (x1)) . . . δ(iJ (xn))

∣∣∣∣J =0

= 〈0|Tφ(x1) . . .φ(xn) exp i(∫

d x4L(φ(x), φ(x))|0〉 (3.9)

with the time-ordering operator T . The functional W generates sums over all connectedgraphs and is defined using the fact that Z[0] gives the sum of all vacuum graphs as

Z = exp(iW) . (3.10)

As one further step it is possible to write down the generating functional Γ for thesum over one-particle irreducible graphs by Legendre transformation. For this the vacuumexpectation value of φ in presence of sources

〈φ〉J =δWδJ

(3.11)

is used and 〈φ〉J is usually called φ(J )cl . Then, the effective action Γ is defined as

Γ[φcl ] = W[J ] −∫

J φcl . (3.12)

The source J is given as

J = −δΓ

δφcl. (3.13)

Hence, the effective action Γ is the classical action and the sum over quantum corrections.It can be used to find the vacuum of the full theory as described in the following.

3.1.3. EFFECTIVE POTENTIAL

The effective potential is the zero momentum limit of the effective action. It is derived byusing a constant scalar field φcl (x) = φ0. The effective action Γ[φ0] then contains a constantspace-time volume factor V4 from the space-time integral and the effective potential V (φ0)is defined as

Γ[φ0] = −V4V (φ0) . (3.14)

From eq. (3.13) it can be seen that for vanishing sources the effective action has to satisfy

δΓ

δφcl (x)= 0 . (3.15)

27

3. Input parameters for the MRSSM

This gives the conditions for the possible values of φcl similar to a classical equation of mo-tion but also including quantum effects. When the definition of the effective potential (3.14)is inserted into eq. (3.15) for constant φ0, it reduces to

∂V∂φ0

= 0 . (3.16)

The global minimum of the effective potential then is the state with the lowest energy andhence gives the true vacuum of the quantum field theory including quantum effects [100].

As the effective potential is defined via the zero momentum limit of the effective actionΓ, it can also be used to calculate one-particle irreducible graphs with vanishing externalmomenta. For example, two-point functions with zero external momenta are given bythe second derivative of the effective potential. This is used in the next chapter to deriveanalytical expressions for one-loop corrections to the Higgs boson mass.

3.1.4. RENORMALIZATION GROUP EQUATIONS

The process of regularization and renormalization in a quantum field theory introduced anadditional dimensionful parameter Q. From a physics point of view, observables shouldnot depend on Q as it is an artifact of the calculation. This is true when considering allorders in perturbation theory, but in a calculation up to a specific order logarithms of theform log E/ Q appear, with E being a typical energy scale of the considered process. Theselogarithms can become large such that perturbation theory breaks down. Therefore, animproved renormalization procedure is required.

The renormalized parameters of a theory depend on Q. The use of the renormalizationgroup allows to resum such possibly large logarithms to all orders consistently. Therenormalization group equation (RGE) of a parameter λ is

∂∂ log Q

λ(Q) = βλ , (3.17)

where the so-called β function βλ, which describes the running of λ with the scale Q, isderived in praxis from the UV divergent part of loop diagrams. If a theory contains severalparameters, there is a β function for each possibly depending on all coupling and thereforea coupled system of RGEs needs to be solved. Using RGEs to run the parameters to therelevant energy scale E of the observable considered, it is now possible to effectivelyresum the logarithms.

3.1.5. EFFECTIVE FIELD THEORIES

The usage of RGEs can ensure the validity of the perturbation expansion for observablesand theories with one relevant energy scale. For realistic theories like the SM or theMRSSM, there are usually several scales present in a calculation. Especially if the theorycontains states i with different masses Mi much above the scale E of the process ofinterest, possibly large logarithms of the form log Mi / E appear. The effective field theory(EFT) approach handles these problematic large ratios by introducing two regimes aboveand below a matching scale where the used theories are different. Above the scale the fulltheory including the heavy modes is specified. Below, the effective field theory is used,which does not contain the heavy states. The connection between the theories is found

28

3.2. Application to the MRSSM

via matching the physics of both theories at the scale separating the regimes. For this, itis used that S-matrix elements in both regimes with light states as external fields needto give the same result. This yields matching conditions between the parameters of thedifferent regimes. Then, the different renormalization of the theories leads to differentRGEs, resumming the problematic logarithms [101]. Alternatively, the EFT approach can beinterpreted as taking the generating function Z (3.8) of the full Lagrangian density includingthe heavy states and integrating out the states not appearing in the low-energy regime.This gives rise to an effective Lagrangian density with additional contact interactions andrelations to the full Lagrangian density. The EFT approach is used in this work by matchingparameters of the SM to the MRSSM. The framework for this is described in the followingsection.

3.2. APPLICATION TO THE MRSSM

The MRSSM needs to be connected to experiment to test it as description of nature. Thisrequires a full quantum field theoretical treatment using the methods described before. Inthe following, the renormalization of the MRSSM and matching to the SM are described indetail. They are presented as basis of a mass spectrum generator for the MRSSM.

A mass spectrum generator is a computational tool using the calculations describedbefore in a computer code to go from the model definition and experimental inputs togiving the complete mass spectrum of the model as output for predictions and calculationsof further observables. The implementation of a mass spectrum generator is described insection 3.3. The calculation for the MSSM can be found e.g. in ref. [102]. Implementationsof spectrum generators as described in this paper for SUSY models are manifold [103–109].

3.2.1. RENORMALIZATION OF THE MRSSM

The renormalizability of the MSSM was proven in ref. [110]. No such proof exists yet forthe MRSSM but as it exhibits all the features of a renormalizable model it is assumed assuch for this work. For the matching to the SM and the calculation of the mass spectrumthe renormalization of the MRSSM is as follows:

• The MRSSM is considered to be CP-conserving. The CP violation originating fromthe quark sector [76] is not considered in this work. Also, as flavor observables arenot of interest, the CKM matrix is assumed to be diagonal. The Feynman gauge ischosen.

• The parameters m2Hu

, m2Hd

, vS and vT are exchanged for the tadpoles tu, td , tS and tT

of the scalar potential as free parameters.• The tadpoles are renormalized such that they minimize the loop-corrected effective

potential up to the considered order. Then, the vev renormalization of refs. [111,112] applies. In the following, this includes one-loop corrections. When the precisecalculation of the CP-even Higgs scalar mass is considered also two-loop effects aretaken into account using the effective potential.

• All other free parameters are chosen to be renormalized in the DR scheme. This isdone implicitly and no counterterms are calculated as described in section 3.1.1.

With this prescription the meaning of all parameters in the MRSSM is fixed and as a nextstep the connection to experiment and the SM is made.

29

3. Input parameters for the MRSSM

3.2.2. MATCHING TO THE SM

After the discovery of the Higgs boson and its subsequent mass measurement, all parame-ters of the Standard Model can be fixed from experimental data at the LHC. This allows fora simple connection of any model extending the SM to experiment. The values of requiredobservables are given in several references, e.g. from the Particle Data Group [98]. Hence,a corresponding set of parameters in the MRSSM can be fixed by matching conditions withthe SM. There exist several possible sets of observables, in this work the experimentalinput observables used are

• the pole mass of the Z boson mZ ,• the MS renormalized fine-structure constant taken at the Z mass pole αMS

SM(mZ ),• the muon decay constant Gµ,• the five flavor MS renormalized QCD coupling constant αMS

S,SM(mZ ) ,• the pole mass of the top quark mt and the τ , µ and electron lepton,• MS renormalized masses of the other quarks.

The values for these observables used in this work are given in the Appendix A.2. Ingeneral, also the pole mass of the Higgs boson mH might be used to fix a parameter ofthe MRSSM. As it is a prediction of a realistic SUSY model and dependent on most of thenew model parameters in a non-trivial way, this is not done here. Rather the Higgs bosonmass is calculated and the experimental value is used to constrain the parameter space inconjunction also with other observables.

With the matching procedure from above the SM observables are used to fix thefollowing DR renormalized parameters of the model: The three gauge couplings g1, g2, g3,the vev v =

√vd

2 + vu2 and the Yukawa coupling matrices Yd , Yu, Y`. (as CKM mixing is

neglected these matrices are all assumed to be diagonal) appearing in the superpotential.The matching condition for the DR renormalized fine-structure constant is

αDRMRSSM(mZ ) =

αMSSM(mZ )

1 − ∆αDR(mZ ),

2πα∆αDR(mZ ) =

13

−6∑

i=1

(13

logml±i

mZ+

19

logmdi

mZ+

49

logmui

mZ

)−

169

logmt

mZ

−3∑

i=1

13

logmH±i

mZ−

2∑i=1

13

logmR±i

mZ−

2∑i=1

43

(log

mχ±i

mZ+ log

mρ±i

mZ

) (3.18)

with α the electromagnetic coupling in the Thomson limit. For the DR renormalized QCDcoupling constant the correction is

αDRS,MRSSM(mZ ) =

αMSS,SM(mZ )

1 − ∆αDRS (mZ )

,

2π

αMSS,SM(mZ )

∆αDRS (mZ ) =

12

−23

logmt

mZ−

6∑i=1

16

(log

mui

mZ+ log

mdi

mZ

)−

12

logmσO

mZ−

12

logmφO

mZ.

(3.19)

These translation rules follow ref. [113]. Even as αMSSM and αMS

S,SM are derived as genuine SM

30

3.3. Implementation

parameters, they do not receive contributions from additional charge or color neutral fieldswith masses below MZ in an extended theory. This allows for such light particles to appearin the MRSSM and in the last part of this chapter a scenario, where new light states like alight neutral Higgs boson appear, is discussed.

With the choice of input parameters discussed before, the pole mass of the W bosonbecomes a prediction of the calculation. This is discussed in detail in chapter 5. Inthe process of this calculation also an expression for the weak mixing angle tan θW =g1/ g2 is derived (see eq. (5.5)), which in combination with αDR

MRSSM allows to calculate theelectroweak gauge couplings:

e =√

4παDRMRSSM , g1 =

e

cos θ, g2 =

e

sin θ. (3.20)

The vev v is calculated from the Z boson pole mass by removing the one-loop correctionsfollowing eq. (3.7) and using the tree-level mass relation (2.59). The Yukawa couplings areextracted in a similar fashion from the pole masses of the SM fermions by subtractingthe loop corrections and using the tree-level relations or deriving them using the MS inputmasses.

3.2.3. POLE MASS PREDICTION FOR SUSY STATES

When the chosen set of parameters is matched to the SM, the two-loop MRSSM renor-malization group equations, given in the appendix B, are used to run the parameters tothe SUSY scale. All parameters that are not fixed by matching to the SM are then givenat this scale and the pole masses of the supersymmetric particles are derived. For thisthe DR renormalized self-energies of all SUSY fields are calculated and the root of theinverse propagator is determined. All the relations used at the electroweak scale rely onthe masses calculated at the SUSY scale. Hence, for a consistent calculation this matchingneeds to be done in an iterative approach by running between the scales and recalculatingthe masses until convergence is reached. In the next chapter, the procedure is describedfor the SM-like Higgs boson as loop corrections play a crucial role for a realistic predictionfor its mass. In the remainder of this chapter the implementation of the mass spectrumcalculation and different promising scenarios in the MRSSM are discussed.

3.3. IMPLEMENTATION

The algorithm to derive the mass spectrum of a SUSY model like the MRSSM described inthe previous sections is implemented in a computer code using the Mathematica packageSARAH-4.8.5 [114–118]. The program produces source files for the spectrum generatorSPheno-3.3.8 [106, 119] which then is enabled to calculate the mass spectrum for giveninput parameters using the SLHA-2 [120, 121] format. The renormalization scale is chosento be Q = 1 TeV and tanβ as input given also at this scale following SPA convention [113].Additionally, the program FlexibleSUSY-1.2 [122, 123] exists that also relies on SARAHbut uses different approaches at several points of the calculation. At the final stage acomparison of the results for the mass spectrum up to a common accuracy betweenSARAH/ SPheno and FlexibleSUSY is possible and an important check of the calculation.SARAH can also produce a model file with tree-level vertices for the Mathematica packageFeynArts-3.7 [124]. Together with the package FormCalc-8.4 [125, 126], it allows to

31

3. Input parameters for the MRSSM

calculate tree-level and one-loop matrix elements in the MRSSM analytically. During theresearch culminating in this thesis, the available diversity of calculation tools was usedextensively to cross check the consistency of the results given by the respective programs.It allowed to give constructive feedback to the authors of the codes leading to an improvedagreement between them and also the literature. Improvements in the tools from providedfeedback include the following:

• The model files for the MRSSM in SARAH were adapted, so that it is possible inSPheno to use the full matching of the MRSSM to the SM instead of a providedsimplified version.

• The contribution of the triplet vev vt SARAH/ SPheno to the W boson mass calculationwas implemented as is described in chapter 5, specially considering the parameter ρdefined therein.

• A more accurate agreement between SARAH/ SPheno and FlexibleSUSY implemen-tation for the mass prediction of the non-MSSM-like states, the R-Higgs bosons, thescalar and pseudoscalar octet was achieved. For this the different numerical resultsand codes were compared and unphysical differences identified.

• The implementation of two-loop corrections to the Higgs boson mass described inchapter 4 was tested extensively.

An overview of all software packages used for acquiring the results of this thesis is givenin the Appendix A.2. The one- and two-loop RGEs are given in Appendix B. Appendix Csummarizes the mixing of the gauge eigenstates to the mass eigenstates and contains thetree-level mass matrices of the SUSY states.

3.3.1. COMPARISON TO EXPERIMENT

With the calculation of loop-corrected masses and couplings it is possible to make theconnection to experiment. The most straight forward comparison is possible for polemass predictions, as they are a direct result of the mass spectrum generator. This isdone in chapter 4 and 5 for the Higgs and W boson mass, respectively. It is clear thatagreement needs to be reached within the respective uncertainty bands of the predictionand measurement so that the MRSSM is a good description of nature. The dark matter relicdensity prediction of the MRSSM is compared to the result from the Planck experimentin a similar fashion. Details on the calculation are given in chapter 6. For observablesrelated to the scalar discovered at the LHC the approach is similar. A set of measurementsof Higgs properties, called the modified signal strengths µi , is used. They measure themodification of Higgs boson production and decay into a certain final state relative to theSM. Comparing the model predictions against the experimental measurements allowsfor a χ2 test from which a p-value can be calculated, expressing the compatibility of theparameter point with data.

The situation is more involved when considering direct searches for BSM physics withnull results. Then, exclusion limits need to be set in the parameter space of the MRSSM.This is done by testing for each parameter point the predicted number of signal events ns

and SM background events nb against the number of observed events n. The cross sectionprediction of the MRSSM for the relevant processes is the model input to calculate thesignal event rate while background and observed rates are taken from the experimentalcollaborations. Using these numbers it is possible to calculate confidence level CLs+b for a

32

3.4. Scenarios in the MRSSM

specific search. For the LHC studies the quantity CLs is used [127]. For each parameterpoint of the model and if no significant signal above the background is observed, exclusionlimits can be derived. A point is assumed to be excluded by 95% (90%) confidence level(C.L.) if CLs+b < 0.05 (CLs+b < 0.1). Details on the calculation of confidence levels from ns,nb and n can be found in refs. [98, 128]. The BSM searches studied in this thesis includethe searches for an extended Higgs sector at LEP, Tevatron and LHC described in chapter 4.Furthermore, the dark matter direct search result by the LUX experiment discussed inchapter 6 and LHC for electroweak SUSY searches discussed in chapter 7 are examinedusing these methods.

A possible next step to investigate the parameter space of the model is the approach ofglobal fitting. This has been done for certain MSSM parametrizations [129–133]. For thisapproach the experimental and predicted values for observables as well as the confidencelevels from exclusion limits are converted into χ2 values and summed, assuming thatcertain requisites are met. Then, it is possible to look for the global best-fit-point in therelevant parameter space. This procedure is not used for the MRSSM in this thesis.Here, the main goal is to investigate the parameter dependencies of the above mentionedobservables and identify relevant parameter regions in the highly-dimensionful parameterspace. Applying global fitting remains as a possible endeavor for future work.

3.4. SCENARIOS IN THE MRSSM

The MRSSM features a rich particle content introduced in the last chapter. For a meaningfulinvestigation of its physics it is helpful to introduce benchmark scenarios that representinteresting regions of parameter space and contain a rich phenomenology at the TeV scale.This fixes the input parameters of the model which can not be derived from experiment.

The general feature of viable parameter points should be the prediction of an SM-likephysics below the TeV scale, as the SM is a very good description of nature at such energies.It could be assumed that all BSM states are rather heavy, therefore not directly accessibleby previous experiments and influencing precision observables through quantum effectsonly within the uncertainty bands. This scenario of the MRSSM is labeled the normalordered scenario in this work and is specified in more detail below.

The Next-to MSSM (NMSSM) is the smallest non-trivial extension of the MSSM byintroducing an additional superfield which is a singlet of all gauge groups.* Excitingly, it isexperimentally not excluded that the scalar component of the singlet superfield could belighter than the SM-like Higgs boson discovered at the LHC [136–143]. The MRSSM alsoincludes a singlet superfield to allow for a Dirac mass term of the bino. It is of high interestto study the phenomenology of the parameter space where this state provides the lightestHiggs boson similarly to the NMSSM references. This scenario of the MRSSM is calledthe light singlet scenario in this thesis and is discussed after the normal ordered scenario.

3.4.1. NORMAL ORDERED SCENARIO

The normal ordered scenario has a mass hierarchy similar to the MSSM. In tab. 3.1 threebenchmark points are defined for the normal ordered scenario. Each represents a differentrange in tanβ. The benchmark points are the benchmark points BMP1’, BMP2’ and BMP3’

*See refs. [134, 135] for reviews of the NMSSM.

33

3. Input parameters for the MRSSM

BMP1 BMP2 BMP3

tanβ 3 10 40Bµ 5002 3002 2002

λd , λu 1.0, −0.8 1.1, −1.1 0.15, −0.15Λd , Λu −1.0, −1.11 −1.0, −0.85 −1.0, −1.03MD

B 600 1000 250m2

Ru20002 10002 10002

µd , µu 400, 400MD

W 500MD

O 1500m2

T , m2S, m2

O 30002, 20002, 10002

m2q;1,2, m2

q;3 25002, 10002

m2dR ;1,2

, m2dR ;3

25002, 10002

m2uR ;1,2, m2

uR ;3 25002, 10002

m2` , m2

eR10002

m2Rd

7002

vS 5.3 1.1 −0.22vT −0.28 −0.05 −0.21m2

hd6742 7642 11622

m2hu

−4982 −5052 −5052

Table 3.1. Benchmark points of the normal ordered scenario. Dimensionful parametersare given in GeV or GeV2, as appropriate. The first part gives input parameters,with values specific for each point, the second, where the values are set to acommon number for all points. The last part shows the derived parameters.

from ref. [144]. They include updated values compared to ref. [145] and within this thesisthe dashes are drop to simplify notation. They all feature rather small fermionic Dirac massparameters, except for a gluino one, and heavy scalar mass parameters especially m2

T ,which is needed to ensure a small vT via the tadpole equations. Values of the superpotentialparameters λ and Λ are generally chosen in magnitude close to unity as is required toachieve the correct SM-like Higgs boson mass, shown in chapter 4.

The mass spectra resulting from the chosen set of parameters is given on the left panelsof fig. 3.1. All new states are heavier than the SM-Higgs boson, denoted as H1. Thecharginos and neutralinos are light enough to be accessible by the later phases of the LHCas are the third generation squarks and pseudoscalar octet. The other scalar fields arerather heavy but affect the phenomenology of the MRSSM indirectly through effects onthe Higgs boson mass and the electroweak precision observables.

3.4.2. LIGHT SINGLET SCENARIO

The light singlet scenario features additional SUSY states lighter than the SM-like Higgsboson. In tab. 3.2 three benchmark points are defined for the light singlet scenario. Thebenchmark points are taken from ref. [146]. They represent different regions in tanβand hierarchies in MD

B and m2S, which are smaller than the SM Higgs boson mass. Each

benchmark point features small λ parameters due to constraints from Higgs observables

34

3.4. Scenarios in the MRSSM

BMP4 BMP5 BMP6

tanβ 40 20 6Bµ 2002 2002 5002

λd , λu 0.01, −0.01 0.0, −0.01 0.0, 0.0Λd , Λu −1, −1.2 −1, −1.15 −1, −1.2

MDB 50 44 30

m2S 302 402 802

m2Ru

, m2Rd

10002, 7002

µd , µu 130, 650 400, 550 550, 550MD

W 600 500 400MD

O 1500m2

T , m2O 30002, 10002

m2q;1,2, m2

q;3 15002, 7002 13002, 7002 14002, 7002

m2dr ;1,2

, m2dR ;3

15002, 10002 13002, 10002 14002, 10002

m2ur ;1,2, m2

uR ;3 15002, 7002 13002, 7002 14002, 7002

m2˜;1,2

, m2eR ;1,2 8002, 8002 10002, 10002 5002, 3502

m2˜;3

, m2eR ;3 8002, 1362 10002, 10002 5002, 952

vS −58.6 −56.6 −56.1vT −0.97 −0.87 −1.1mhd 12152 7302 10422

mhu −(7692) −(6612) −(636)2

Table 3.2. Benchmark points for the light singlet scenario. Layout is as for tab. 3.1.

and Λ in the region of unity to provide the correct SM Higgs boson mass; both requirementsare derived in the next chapter. As with the normal ordered scenario the Dirac massparameters are small and m2

T is large to ensure the smallness of vT . The other scalarmasses are still larger than the fermionic mass parameters but lower than in the normalordered scenario.

The mass spectra resulting from the chosen set of parameters is given on the rightpanels of fig. 3.1. The masses of the fields are generally lighter than in the normal orderedscenario. This is of course most significant for the singlet-like Higgs boson (H1) whichis required to be lighter than the SM-like Higgs boson, denoted here as H2. The lightestneutralino has a mass below half the SM-like Higgs boson mass as derived in the nextchapter. This low mass makes the lightest neutralinos the lightest supersymmetric state.It is stable due to the R-symmetry and the dark matter candidate of the model. The otherneutralinos and charginos are also rather light and in reach of the LHC. Also, dark matterconstraints may require relatively light sleptons, especially staus, also accessible by theLHC. The squarks of the third generation are lighter than in the normal ordered scenario.

35

3. Input parameters for the MRSSM

0

400

800

1200

1600

2000

2400

2800

3200

3600

Mas

s/

GeV

H1

A1H2R1

H±1

H3

A2R2

H±2

H4A3

H±3

OPt, b

q

ν, l±τ±

R

g

χ1χ3

χ2 χ±1

χ±2

χ4ρ±1ρ±2

OS

0

400

800

1200

1600

2000

2400

2800

3200

3600

Mas

s/

GeV

A1H1H2

H±1A2

H3

R2

H±2

H4A3

H±3

R1 t, b

qg

τ±R

ν, l±

χ1χ2 χ±1

χ3χ4 χ±

2ρ±1ρ±2

OS

OP

0

400

800

1200

1600

2000

2400

2800

3200

3600

Mas

s/

GeV

H1

R1A1H2

H±1

H3

A2

H±2

H4A3

H±3

R2OPt, b

q

ν, l±τ±

R

g

χ1χ3

χ2 χ±1

χ±2

χ4

ρ±1ρ±2

OS

0

400

800

1200

1600

2000

2400

2800

3200

Mas

s/

GeV

A1H1H2

H±1A2

H3R1

H±2A3

H4 H±3

R2

t, b

q

τ±R

ν, l±

g

χ1

χ2 χ±1

χ±2χ3

χ4ρ±1ρ±2

OS

OP

0

400

800

1200

1600

2000

2400

2800

3200

3600

Mas

s/

GeV

H1

R2A1H2

H±1

A2H3

H±2A3

H4 H±3

R1 OPt, b

q

τ±R

ν, l±

g

χ1

χ2χ4

χ±1

χ±2χ3ρ±1ρ±2

OS

0

400

800

1200

1600

2000

2400

2800

3200

Mas

s/

GeV

A1H1H2

H±1R2

A2H3

H±2A3

H4 H±3

R1t, b

q

τ±R

ν, l±

g

χ1

χ2 χ±1

χ±2

χ3χ4ρ±1ρ±2

OS

OP

Figure 3.1. Shown are the mass spectrum plots of the Benchmark points. The left panelsgive the spectra for normal ordered BMP1 to BMP3 from top to bottom. Theright plots give them for the light singlet BMP4 to BMP6 from top to bottom.The plots were generated using PySLHA [147].

3.5. SUMMARY

In this chapter, the foundation for the study of the MRSSM has been set. The requiredquantum field theoretical concepts and their respective implementation have been de-scribed above. With the interesting parameter regions of the MRSSM identified in the lastsection, it is possible to study the phenomenology of the model extensively and derivepredictions for experiment. This is done for the Higgs sector in chapter 4. The impactof the MRSSM states on electroweak precision observables is the focus of chapter 5.Comparison to results from dark matter experiments and direct searches for electroweakBSM states at the LHC are done in chapter 6 and 7, respectively. The findings are thencombined in chapter 8.

36

4. SUSY HIGGS PHYSICS

With the discovery of an SM-like Higgs boson with a mass of

mexpH = 125.09± 0.24 GeV (4.1)

at the LHC [148–154], the Standard Model is seemingly complete. A light scalar with amass of at most 135 GeV or below is also a prediction of the MSSM if the other SUSYstates have masses at the TeV scale [155–157]. It is of interest how the discovered bosoncan be explained in the context of the MRSSM and how the predictions for its interactionscompare to the SM and to experiment.

For the MSSM it is well known that loop effects play an important role to achievethis mass. Especially stops, the superpartners of the top quarks, and their mixing yieldsubstantial contributions to the Higgs boson mass in the MSSM. In the MRSSM mixingbetween the left- and right-handed stops is forbidden by R-symmetry and additionalcorrections are required. Moreover, the Higgs sector of the MRSSM is larger than in theMSSM leading to interesting features, especially considering the singlet-like state.

In the last chapter, two interesting parameter regimes in the MRSSM have been in-troduced. In the normal ordered scenario the SM-like Higgs boson is the lightest scalarstate. This parameter region is the basis for the initial study of the Higgs physics of theMRSSM. The first part of this chapter elaborates on the mechanism to achieve these newcontributions. There, the relevant quantities on tree level are introduced and additionalone-loop corrections are outlined, first using the effective potential approximation andafterwards also taking into account the full one-loop contributions. In a second step thetwo-loop corrections from the effective potential approximation are added and discussed.The last part of this chapter deals with the peculiarities of the second scenario described inthe previous chapter, where the lightest Higgs boson is singlet-like and the second lightestis the SM-like Higgs boson. Then, the mass hierarchy allows for compelling predictionsof Higgs observables. Special emphasis is put on the mixing of the lightest state withthe SM-like Higgs boson and the effects on the phenomenology of Higgs observables.Additionally, constraints from this scenario can be directly applied to dark matter and directLHC searches, which are discussed in latter chapters of this work.

The results concerning the Higgs boson mass calculation at one and two loops are basedon refs. [144, 145]. The light singlet scenario was first investigated in ref. [146].

37

4. SUSY Higgs physics

4.1. THE HIGGS SECTOR OF THE MRSSM

After electroweak symmetry breaking, the MRSSM Higgs sector contains four scalar, threepseudo-scalar and three charged Higgs fields. The mass matrices at tree-level for thesethree sectors are given by eqs. (C.5) to (C.7) of the appendix. When the solutions to thetadpole eqs. (C.4) are inserted, the φd -φu sub-matrices correspond to the mass matrices ofthe MSSM Higgs fields.

For the following discussion on quantitative features a certain mass hierarchy betweenthe different gauge eigenstates is assumed; no such assumptions are included whenpresenting quantitative results. The SM-like Higgs boson seen at the LHC is taken as thelightest of the two MSSM doublet scalars. The MSSM scenario where it is the heavierstate is not completely excluded but strongly constrained [158–161]. The triplet states arethe heaviest fields as electroweak precision observables (see chapter 5) put strong boundson the triplet vev vT , which is related to the triplet soft breaking mass m2

T by means ofthe tadpole equation (C.3). The requirement of a large scalar mass is sufficient to achieveacceptable suppressed values for vT . The singlet states may be heavier or lighter than theSM-like Higgs boson leading to very different experimental signatures. For the followinganalytical derivations it is assumed that the singlet state is heavier than the SM-like Higgsboson; modifications to the phenomenology for the opposite case are discussed in the lastpart of this chapter.

In the MSSM, an upper limit on the tree-level mass of the lightest doublet scalar statecan be derived. In the decoupling limit, where mA mZ and the mixing angle α of thescalar Higgs boson mass matrix MH,MSSM goes to α = β − π/ 2, this limit takes a clear formas

m2H1,approx, max = m2

Z cos2 2β . (4.2)

In the MRSSM, the case is more problematic as the mass matrix is a 4 × 4 matrix andthe additional mixing with the singlet and triplet states drives the lowest eigenvalue of themass matrix to lower values. This can be seen when a similar limit as in the MSSM is takenwith mA mZ and additionally λ = λu = −λd , Λ = Λu = Λd , µu = µd = µ, and vS ≈ vT ≈ 0is used. Then, the result in eq. (4.2) becomes

m2H1,approx, max = m2

Z cos2 2β − v2

(

g1MDB +√

2λµ)2

4(MDB )2 + m2

S

+

(g2MD

W + Λµ)2

4(MDW )2 + m2

T

cos2 2β .

(4.3)Clearly, the result in neither the MSSM nor the MRSSM is in agreement with the ex-perimental data of mZ = 90.2 GeV and the Higgs boson mass of eq. 4.1. It is a knownfact [162–166] that radiative corrections to the Higgs boson mass can be very large in SUSYtheories and need to be taken into account for a realistic prediction. They are responsiblefor lifting the Higgs boson mass above the mass of the Z boson as seen by experiment. Inthe following, it is shown how this is possible using the additional field content and thenew couplings of the MRSSM.

38

4.2. Quantum corrections to the CP-even Higgs boson mass

4.2. QUANTUM CORRECTIONS TO THE CP-EVEN HIGGS BOSONMASS

The pole mass of the Higgs field needs to be calculated taking into account quantumeffects to the self-energy (Σi j (p2). Then, the roots of the inverse full propagator have to befound at the pole mass:

0 != det[p2δi j − m2

i j + <(Σi j (p2))]

p2=m2pole

, (4.4)

where p is the momentum, m2i j the tree-level mass matrix with renormalized parameters

and Σi j (p2) the DR renormalized self-energy matrix of the scalar. There are several ap-proaches to find the solutions to this equation. They differ in the achievable accuracyof the mass prediction and speed with which the computation is done. This hinges onthe treatment of the momentum dependency as for the exact result the pole mass hasto be used as argument of the self-energy. Two different approaches are discussed inthe following. Firstly, the Feynman-diagrammatic calculation of the self-energies keepingthe momentum dependency is introduced. Secondly, the effective potential approachis described. Here, the momentum is assumed to be small compared to the massesappearing in the self-energy and therefore neglected.

4.2.1. DIAGRAMMATIC APPROACH

The one-particle-irreducible self-energy of the Higgs boson is derived by calculating allcontributing Feynman diagrams up to the required loop order. For the one-loop levelcontributing diagrams are shown in fig. 4.1. The generic fermionic and bosonic fieldsshown there have to be replaced by the concrete fields appearing in the MRSSM. Theresult can then be inserted into eq. (4.4) and used to find the pole mass of the Higgs boson.

S′

S′

S S S SS′

Figure 4.1. Generic scalar self energies

For a calculation at a certain loop order it is within the accuracy of the perturbation seriesto use a mass derived at a lower order as momentum argument of the self-energy. Forexample, it is enough for an one-loop calculation to insert tree-level masses in the Feynmandiagrams. Still, as the Higgs boson mass is especially sensitive to corrections from theSUSY scale, a precise prediction is required and to achieve the highest possible precision,an iterative procedure using the full pole mass is done. The tree-level mass is used for thefirst step while for the consecutive iterations the mass value calculated one step before isinserted until convergence is reached.

The resulting mass matrix is momentum dependent and needs to be diagonalized suchthat the momentum corresponds to the considered eigenvalue. Hence, a 4 × 4 massmatrix needs to be diagonalized four times so that for each mass the correct momentumis inserted. Each diagonalization yields a separate mixing matrix for different momenta.As unitarity needs to be preserved, only one of them can be used in calculations and thechoice is ambiguous. Usually the one belonging to the lightest eigenvalue is chosen.

39

4. SUSY Higgs physics

4.2.2. EFFECTIVE POTENTIAL

The diagrammatic calculation gives the most accurate result as it includes all necessarycontributions. Even so, the expressions become rather involved and it is not straightforwardto grasp an immediate understanding from them. The effective potential approach is ahelpful alternative, which provides the Higgs boson self-energy at zero momentum andcan be used as approximation in eq. (4.4). The basic definition of the effective potentialhas been given in section 3.1.3. It has the useful feature that the self-energy Σ(0) of theHiggs bosons can be obtained from the second derivatives of the effective potential at zeromomentum. For illustration, the effective potential in softly broken supersymmetry at theone-loop level has the compact form [167]

V 1Leff =

164π2

∑i

(−1)2Si+1(2Si + 1) Tr

[M4

i

(log

M2i

Q2 −32

)], (4.5)

summing over all fields with Higgs field dependent mass matrix M2i and spin Si in the DR

scheme and Landau gauge. Q is the DR renormalization scale.Taking the appropriate derivative of the effective potential (4.5), the gauge-independent

loop contributions to the Higgs boson masses can be approximated by the result, as all theconsidered parameters needed in eq. (4.5) are available in the DR scheme. Compared tothe full calculation as described in section 3.3, which is done using the Feynman gauge, theexpression (4.5) is valid in Landau gauge. To avoid problems stemming from this difference,the gauge-less limit is chosen when comparing results. For this, the gauge couplings of thebroken gauge group SU(2)L × U(1)Y are set to zero. This neglects especially contributionsfrom massive gauge and Goldstone bosons but the result is valid for both gauges.

When the p2 dependence of the self-energy is neglected, eq. (4.4) can be solved directlyfor the one-loop corrected mass matrix as

m2pole, approx;i j = m2

i j +∂2V 1L

eff

∂φi∂φj

∣∣∣∣∣φi=0,φj=0

. (4.6)

Diagonalizing this mass matrix provides the Higgs boson masses without a momentumdependency and, therefore, one mixing matrix can be found unambiguously.

A possible problem of the effective potential approach is the so-called “Goldstonecatastrophe” which is described and treated in refs. [168–170]. It is based on the factthat the DR renormalized Goldstone boson masses appearing in the loop expressions canhave any value for zero on-shell masses. If these masses approach zero the tadpolesderived from the effective potential diverge. A possible solution is the resummation of theGoldstone contributions.

4.2.3. ONE-LOOP CORRECTIONS

In the following sections the methods introduced before are used to derive predictions forthe pole mass of the Higgs boson* at the one-loop level and to present the dependence onthe model’s parameters.

*In the following, Higgs boson (pole) mass generally is used to denote the mass of the SM-like Higgs field inthe model.

40

4.2. Quantum corrections to the CP-even Higgs boson mass

At first, an analytical formula from the effective potential approach is discussed and thencompared to the full result from the full Feynman diagrammatic calculation.

ANALYTICAL RESULTS

As an illustrative example the same limit as for eq. (4.3) is used, where λ = λu = −λd ,Λ = Λu = Λd and vS ≈ vT ≈ 0 as well as assuming large mA and tanβ. Then, thelightest Higgs state is given mainly by the φu component and the (φu,φu)-component ofthe mass matrix in eq. (C.6) which can be treated as the dominant contribution. ExpandingV 1L

eff (4.5) in powers of φu, where terms of higher order than O(φ4u) are suppressed by

denominators containing m2S and m2

T , simple analytical terms can be found. The mostimportant contributions are the ones in fourth order of λ and Λ.

The one-loop corrections to the SM-like Higgs boson mass proportional to Λ and λ thenare

∆m2H1,eff.pot,λ =

2v2

16π2

[Λ2λ2

2+

4λ4 + 4λ2Λ2 + 5Λ4

8log

m2Ru

Q2

+

(λ4

2−λ2Λ2

2m2

S

m2T − m2

S

)log

m2S

Q2

+

(58Λ4 +

λ2Λ2

2m2

T

m2T − m2

S

)log

m2T

Q2

−

(54Λ4 − λ2Λ2 (MD

W )2

(MDB )2 − (MD

W )2

)log

(MDW )2

Q2

−

(λ4 + λ2Λ2 (MD

B )2

(MDB )2 − (MD

W )2

)log

(MDB )2

Q2

].

(4.7)

The Q dependencies of the different contributions cancel among themselves. The expres-sion for the contribution from the stop-top fields can be taken from the MSSM in absenceof squark mixing as

∆m2H1,eff.pot,Yt

=6v2

16π2

[Y 4

t log(

mt1mt2

m2t

)], (4.8)

where mt1and mt2

are the masses of the two superpartners to the top quark. Thetop Yukawa coupling Yt is the third diagonal entry of the up Yukawa parameter matrix:Yt ≡ Yu,33. Both have a quartic dependency on the dimensionless parameters, λ and Λ orYt . This may be understood from their common origin as superpotential parameters. Bothcouple the Higgs doublet to another doublet and a singlet (for λ and Yt ) or a triplet (for Λ)of SU(2)L. The difference between singlet and triplet leads to a different prefactor in thecontribution to the Higgs boson mass, as is visible for example from the second term ineq. (4.7).

Hence, a substantial enhancement of the Higgs boson mass from loop effects tocompensate the absence of stop mixing and reduction at tree-level is possible whenthe λ and Λ parameters are chosen to be of order unity, comparable in size to the topquark Yukawa coupling. Even as the tree-level reduction seen in eq. (4.3) is of quadraticorder in the superpotential parameters, the quartic dependency of the loop result (4.7)on them can be enough to generate a mass of the SM-like Higgs boson close to the

41

4. SUSY Higgs physics

experimentally measured value. In the following, this is shown explicitly, taking intoaccount all dependencies as described in section 4.2.1.

NUMERICAL RESULTS

As it has been shown with eq. (4.7), the new λ and Λ parameters of the MRSSM areimportant to achieve a large enhancement to the Higgs mass from loop corrections. In thefollowing, it is discussed which values are needed to get an SM-like Higgs boson mass ofaround 125 GeV. For this, the normal ordered scenario is used where the lightest Higgsboson is SM-like. The results are summarized in figs. 4.2 and 4.3.

Generally, the figures show for each of the four parameters λd , λu, Λd , Λu in the top panelthe tree-level, one-loop and two-loop mass prediction for the SM-like Higgs. The middlepanel shows the mass difference when comparing the full one-loop corrections with theeffective potential calculation using the quantity ∆meff.P. = mH1,full one-loop−mH1,one-loop eff. pot..In the bottom plot the size of the two-loop corrections is depicted, which are discussed inthe next section. Figs. 4.2 and 4.3 are based on the benchmark points BMP1 and BMP2,respectively.

The reduction in eq. (4.3) originating from enlarged mixing of the doublets with the singletand triplet on tree-level can clearly be seen. This reduction is quadratic in the superpotentialparameters. Taking the loop effects from eq. (4.7) with a quartic dependency on theparameters into account compensates the reduction and actually leads to an enhancementof the Higgs boson mass for larger absolute values of the λ and Λ. This can especially beseen for Λu. This is comparable to the top-Yukawa effects as discussed before, which arealso included in the shown result.

The squark masses are chosen to be at one TeV for the benchmark points, and it can beseen that this is not enough to achieve a Higgs boson mass of 125 GeV without the newcontributions. For fig. 4.2 (4.3) the contribution from stop-top loops is 28 GeV (24 GeV).The different values for the benchmark points is due to the different choices of tanβ forthe benchmark points and the resulting difference in the numerical value of the top Yukawacoupling. To further raise the Higgs boson mass, the MRSSM superpotential couplingtypes λ and Λ, especially λu and Λu, are needed to be of order unity.

The difference in tanβ also influences the size of the needed loop contributions. Thetree-level prediction given in eq. (4.3) is reduced for small tanβ, which can be seen bycomparing both benchmark points. Additionally, it affects how important the down-type λd

and Λd are compared to the up-type parameters λu and Λu. It can be seen from the variationof λd and Λd in fig. 4.2 that their respective one-loop corrections contribute at most a fewGeV. For fig. 4.3 these contributions are even less and it is clear that λd and Λd do notcontribute significantly to the Higgs boson mass as they always appear together with thevev vd , which becomes small for rising tanβ. This is why the bulk of the unique MRSSMone-loop contributions stem from the parameters λu and Λu as they are accompanied bythe vev vu.

The difference of the full one-loop calculation to the one-loop effective potential asthe quantity ∆meff.P. is shown in the middle panel of each plot. As this difference isalways negative, the effective potential result is always larger than the full result by severalGeV. The difference is roughly ten percent of the full one-loop correction, showing thatthe effective potential approach is a convenient way to arrive at first results and get anunderstanding of analytical dependencies but the full calculation is needed to minimize the

42

4.2. Quantum corrections to the CP-even Higgs boson mass

60

70

80

90

100

110

120

130

mH

1 [

GeV

]

3.453.403.353.303.253.203.153.103.05

meff

.P. [

GeV

]

1.5 1.0 0.5 0.0 0.5 1.0 1.5

d

4.7

4.8

4.9

5.0

5.1

m2

L [

GeV

]

60

70

80

90

100

110

120

130

mH

1 [

GeV

]

3.30

3.25

3.20

3.15

3.10

3.05

meff

.P. [

GeV

]

1.5 1.0 0.5 0.0 0.5 1.0 1.5

d

4.884.904.924.944.964.985.005.02

m2

L [

GeV

]

020406080

100120140160

mH

1 [

GeV

]

5.0

4.5

4.0

3.5

3.0

2.5

meff

.P. [

GeV

]

1.5 1.0 0.5 0.0 0.5 1.0 1.5

u

3.84.04.24.44.64.85.05.2

m2

L [

GeV

]

0

50

100

150

200m

H1 [

GeV

]

7654321

meff

.P. [

GeV

]

1.5 1.0 0.5 0.0 0.5 1.0 1.5

u

4.55.05.56.06.57.07.5

m2

L [

GeV

]

Figure 4.2. For varying each of the four superpotential parameters λd , λu, Λd and Λu threepanels are shown. All other parameters are chosen as for BMP1. The topplot contains the prediction of the SM-like Higgs boson mass at tree-level(blue, dashed-dotted), one-loop (green, dashed) and two-loop (red, full) level,the middle one the difference of the full one-loop Feynman diagrammaticand effective potential approach and the bottom panel the difference of thetwo-loop and full one-loop result. The lines are not continued when a tachyonicstate is encountered at tree-level mass calculation.

43

4. SUSY Higgs physics

859095

100105110115120125130

mH

1 [

GeV

]

2.770

2.765

2.760

2.755

meff

.P. [

GeV

]

1.5 1.0 0.5 0.0 0.5 1.0 1.5

d

5.005.055.105.155.205.255.305.35

m2

L [

GeV

]

859095

100105110115120125130

mH

1 [

GeV

]

2.7682.7662.7642.7622.7602.7582.7562.7542.752

meff

.P. [

GeV

]

1.5 1.0 0.5 0.0 0.5 1.0 1.5

d

5.105.115.125.135.145.155.165.175.18

m2

L [

GeV

]

020406080

100120140160

mH

1 [

GeV

]

3.23.02.82.62.42.22.01.8

meff

.P. [

GeV

]

1.5 1.0 0.5 0.0 0.5 1.0 1.5

u

4.5

5.0

5.5

6.0

6.5

7.0

m2

L [

GeV

]

20406080

100120140160180200

mH

1 [

GeV

]

7654321

meff

.P. [

GeV

]

1.5 1.0 0.5 0.0 0.5 1.0 1.5

u

56789

101112

m2

L [

GeV

]

Figure 4.3. Shown are the same plots as in fig. 4.2 with the unvaried parameters chosenas for BMP2.

44

4.2. Quantum corrections to the CP-even Higgs boson mass

g

t

t

g

t

t

g

t

t

φO

t

tFigure 4.4. Shown are diagrams contributing to the two-loop effective potential via the

strong coupling.

uncertainties. The difference stems from the missing momentum dependent terms of theeffective potential approach, which are set to zero there.

In this section it has been made clear that the unique MRSSM parameter types λ andΛ need to be of order unity to achieve the correct Higgs boson mass as loop diagramscontaining them contribute with quartic dependence as given by eq. (4.3).

4.2.4. TWO-LOOP CORRECTIONS

The inclusion of one-loop effects is important to arrive at the experimentally measuredHiggs boson mass. It is known from the MSSM that the one-loop result has a theoreticaluncertainty of the order of five GeV. As the experimental value (4.1) has a below GeVuncertainty the inclusion of further loop orders is required for a more precise theoryprediction. Furthermore, new effects may appear on a certain loop order giving rise tointeresting effects.

In the utilized framework of the tools SARAH and SPheno it is also possible to add certaintwo-loop corrections to the one-loop result. These contributions are taken in the gauge-lesslimit, where the gauge couplings g1 and g2 of the broken gauge group SU(2)L × U(1)Yare set to zero circumventing the problem of the “Goldstone catastrophe” discussedbefore. Moreover, the momentum dependency of the two-loop diagrams contributing tothe self-energy Σ in eq. (4.4) is neglected as they are derived in the effective potentialapproach as described in section 4.2.2. All one-loop contributions are continued to be takeninto account completely.

When the two-loop effects are added, two qualitatively different types of contributionsexist. Firstly, contributions that exist at lower loop order get additional corrections. For theMRSSM for example, this is true for contributions from Λ/λ and the Yukawa couplings.Secondly, starting at a certain loop order, also genuine new effects may arise. The strongcoupling constant αs appears for the first time in the self energies for the Higgs bosonmass at the two-loop level. Both effects are investigated in the following.

The two-loop corrections stemming from αs are not only related to the gluon but alsothe Dirac gluino and the sgluon, the scalar component field of the octet superfield O. Thediagrams contributing to the two-loop effective potential are shown in fig. 4.4. From theMSSM it is known that corrections proportional to αs are sizeable of the order of five GeVand also driven by the stop mixing parameter At . See refs. [167, 171–179] for calculationsusing the effective potential approach and refs. [155, 180–182] for Feynman-diagrammaticcalculations.

For the MRSSM stop mixing is absent but the corresponding QCD corrections are stillexpected to be large. Besides the absence of left-right squark mixing, another differenceof the MRSSM to the MSSM is the Dirac nature of the gluino mass parameter MD

O. Itadditionally appears via the mechanism of soft SUSY breaking as coupling parameter of

45

4. SUSY Higgs physics

1000 2000 3000

MDO [GeV]

2

0

2

4

6

8m

2L-m

1L [

GeV

]

all contributions

no sgluon

and no gluino

and no gluon

and no 2t , t b

10002000 3000 4000

MDO [GeV]

5

0

5

10

15

m2

L-m

1L [

GeV

]MRSSM, mO =2 TeV

MRSSM, mO =10 TeV

MRSSM, no sgluon con.

MSSM, no stop mixing

MSSM, stop mixing

1000 2000 3000

MDO [GeV]

5

0

5

10

15

m2

L-m

1L [

GeV

]

MRSSM, mO =2 TeV

MRSSM, mO =10 TeV

MRSSM, no sgluon con.

MSSM, no stop mixing

MSSM, stop mixing

Figure 4.5. The figure compiles the numeric investigation of two-loop contributions to themass of the SM-like Higgs boson depending on MD

O. The left plot shows thecontributions from the different sectors, all other parameters are set as BMP3.The middle (BMP1) and right (BMP3) plot give for the MRSSM two lines forscenarios with different sgluon masses and without sgluon contributions aswell as for the MSSM lines for comparable scenarios with and without stopmixing.

the sgluon to the squarks, especially the stops, via the D term (2.34). Hence, MDO is the

coupling appearing in the vertices of the right diagram in fig. 4.4.

The main contributions from the different sectors to SM-like Higgs boson mass areshown on the left of fig. 4.5 depending on the gluino mass parameter MD

O. The maincorrection comes from the gluon diagram, given by the difference between the red andcyan line, and amounts to around four GeV. The contributions from diagrams containing agluino, the difference of the red and green line, can be around one GeV for large MD

O. Thesgluon gives a positive correction, comparing the blue and green line, which becomes verylarge and of the order of several GeV. For very high MD

O this contribution becomes largerthan the gluon one. The other, non-QCD contributions for the shown points are below halfa GeV.

On the two plots on the right of fig. 4.5 a comparison of the MRSSM to the MSSM isgiven for two different benchmark points. In tab. 4.1 the corresponding relevant MSSMparameters are given which where used to calculate the MSSM lines. It can be seen thatthe MRSSM result excluding the sgluon contributions strongly resembles the MSSM withno stop mixing. This can be understood as similar Feynman diagrams are calculated for theeffective potential. Both of course contain the gluon contribution given by the diagram onthe left of fig. 4.4. As only one chirality of the Dirac gluino couples to quarks, the secondand third diagram also give the same result as in the MSSM.

If the sgluon diagram on the right of fig. 4.5 is included for the MRSSM, the result iscomparable to the MSSM with strong stop mixing. For the MSSM, stop mixing adds an-other contribution to the diagrams. This can be explained in a mass insertion approximation

46

4.2. Quantum corrections to the CP-even Higgs boson mass

tanβ M1 M2 µ mA m2q,u,d;(3,3) m2

q,u,d m2l,e Aτ,b Xt

middle panel 3 600 500 400 700 10002 20002 10002 0 0/2000right panel 40 250 500 400 700 10002 20002 10002 0 0/2000

Table 4.1. Definition of the fixed parameters for the MSSM points in fig. 4.5. All parametersin GeV or GeV2, where appropriate. The stop mixing parameter Xt = At −µ cotβis given both for the case of no and large stop mixing.

where the diagrams then are proportional to the stop mixing parameter Xt , just as theunique MRSSM diagram on the right of fig. 4.5 is proportional to MD

O.As is clear from the left panel of fig. 4.5, the main two-loop corrections stem from the

QCD corrections and amount to around four GeV for the benchmark points. Taking this intoaccount, in figs. 4.2 and 4.3 the two-loop corrected Higgs boson masses varying with theparameters λd , λu, Λd and Λu are shown for BMP1 and BMP2, respectively. The top panelof all plots shows that the two-loop corrections help to bring the Higgs boson mass to theexperimental value of 125 GeV. The bottom panel shows the difference of two-loop andone-loop corrected Higgs boson mass ∆m2L = mH1,2L − mH1,2L. For most of the parameterrange these corrections are around five GeV as in fig. 4.5. Then, the bulk of the contributionis from QCD effects while the contributions from the λ/Λ parameters amount to belowone GeV. Just when the magnitude of Λu and λu rises much above unity the correctionsbecome large. But this is also the regime where the one-loop contributions are sizableand the two-loop diagrams only give a small correction to them. The relevance of Λu

and λu compared to Λd and λd is in line with the discussion on the one-loop corrections,as the parameters always appear together with the vevs vu and vd , respectively, in theexpressions for the self energies, and for large tanβ, vd vu holds.

To summarize, the two-loop contributions to the Higgs boson mass are in the relevantparameter region dominated by the QCD contributions and the effects from the Yukawa-like λ/Λ parameters are sub-dominant and much smaller than their respective one-loopcorrections exhibiting convergence in the perturbation series.

4.2.5. HIGHER-ORDER UNCERTAINTIES AND SCHEME DEPENDENCY

To get an estimate on the uncertainty of the Higgs boson mass from higher-order correc-tions, refs. [183] and [184] for the MSSM and NMSSM, respectively, are used as a startingpoint for the discussion. There, two-loop corrections from the strongly interacting sectorand their implementation in different spectrum generators for the MSSM and NMSSM and,among other things, the influence of different higher-order contributions in the DR schemeare presented. The difference originates from different implementations of the spectrumgenerators that are correct up to two-loop order but differ for higher orders. The two-loopuncertainty in the MSSM and NMSSM for the DR scheme is estimated to be at least 3GeV mainly depending on the treatment of the top Yukawa coupling. The MRSSM sharesthe most important contributions from the strong sector with the other models and, hence,it can be assumed that the uncertainty on the Higgs boson mass from the calculationspresented here is also at least of this magnitude.

Furthermore, all two-loop corrections are derived from the effective potential approachand miss the momentum-depended terms of the full result. To estimate the uncertainty

47

4. SUSY Higgs physics

from this part of the calculation, the relative difference between the full and effectivepotential result at the one-loop level can be used as a measure. As discussed before, thisdifference amounts to around ten percent of the full one-loop correction. Taking this as therelative uncertainty of the two-loop momentum-depended part, it can be assumed that theabsolute contribution is the order of half a GeV.

Another way to understand the uncertainty on the mass is to study the prediction ofdifferent renormalization schemes which also captures differences from higher-order loopcorrections. This is well known for the MSSM [185, 186], where the difference betweenthe on-shell and the DR scheme at the one-loop level can be more than ten GeV. Thisreduces when adding the two loop contributions to three GeV in agreement with theprevious assessment.

Recently, a study for the MRSSM was presented in ref. [187]. There, the results given byfig. 4.5 were confirmed in the DR scheme and extended by comparing to the calculationin the on-shell scheme. It is shown that the result of a steeply rising Higgs boson massfor large values of the gluino mass parameter MD

O should not be interpreted as a physicaleffect but rather an artifact of the DR renormalization, namely the non-decoupling of thescheme.

4.3. HIGGS OBSERVABLES

As the Higgs boson of the SM was long sought after, a wealth of experimental dataexists concerning it and any additional states must evade bounds set by this information.After discovery at the LHC, the time for precision measurements for the properties of theHiggs boson has begun, usually taking the SM Higgs boson as basis. Any BSM model,like the MRSSM, also has to accommodate this data. The tools HiggsBounds-4.2.2 andHiggsSignals-1.4.1 [188–193] take advantage of the available information and provide thepossibility to test a phenomenological model in a correct statistical framework.

The mass spectrum generator of the MRSSM of SARAH and SPheno can produce filescontaining the effective Higgs couplings required as possible input by HiggsBounds andHiggsSignals. Then, the MRSSM Higgs sector of a parameter point is checked against allexperimental constraints included in HiggsBounds by using the option LandH. This includesdata from LEP, Tevatron and LHC, where the mass and signal strength limit from LEPis the strongest quantifier for Higgs bosons below 115 GeV. Above that threshold thehadron collider results are relevant. To verify the properties of the SM-like Higgs bosonall peak observables available are used (option latestresults and peak) with HiggsSignals.The mass uncertainties are described using a Gaussian shape and as discussed beforeit is assumed that the mass of the SM-like Higgs boson has an uncertainty of three GeVfollowing the discussion in the previous section.

A simplified approach is used for the interpretation of the HiggsBounds and HiggsSignalsresults. A parameter point is excluded by HiggsBounds using the 95% C.L. limit given bythe program and by HiggsSignals when its approximately calculated p-value is smallerthan 0.05. When both constraints apply, the point is excluded by both tools. For a morecomplete statistical treatment of extended Higgs sectors see refs. [194] and [195]. In thecase of the MRSSM this approach is left for future work.

Fig. 4.6 shows the first application of HiggsBounds. Here, the exclusion contour inthe plane of the MSSM pseudo-scalar Higgs boson mass mA (2.47) and tanβ is given.

48

4.4. Phenomenology of a light singlet scenario

300 400 500 600 700 800 900 1000

mA [GeV]

10

20

30

40

50

tan

Allowed (yellow)

Excluded by HB (blue)

Figure 4.6. The exclusion limit by HiggsBounds is given in the mA-tanβ parameter plane.For each point Λu has been varied to achieve the correct SM-like Higgs bosonmass. The yellow region is allowed, while the blue region is excluded byHiggsBounds. All other parameters are chosen as for BMP3.

Additionally, the parameter Λu is varied to achieve the correct SM-like Higgs boson massfor each point in the plane. All other parameters are chosen as for BMP3. The plot shows alower limit on the mass mA for each value of tanβ ranging from 300 GeV for tanβ = 5 to1000 GeV for tanβ = 40. An analogous plot is well known for the MSSM, see for exampleref. [196], fig. 1 or 3. There, a wedge-shape region in the comparable parameter spaceis not excluded. When comparing to fig. 4.6 a similar boundary can be found with thesame explanation for both plots. The allowed region gives the parameter space where thedecoupling limit with mA mZ and α = β − π/ 2 (the angle diagonalizing the scalar massmatrix in the MSSM) is valid and the SUSY model can not be separated easily from theSM. For the MSSM, a lower bound of tanβ = 5 independent of mA also exists which doesnot appear for the MRSSM. This bound depends on the scenario of the MSSM, usuallymh

max [197, 198] or mhmod [196], and in this region it is not possible to achieve a lightest

Higgs boson mass above the exclusion limit from LEP [199] of roughly 115 GeV, therebyexcluding this region. This is not the case in the MRSSM, as Λu is adjusted to ensure thecorrect mass to the SM-like Higgs boson.

No figure is given here with regards to HiggsSignals for the normal ordered scenarioas the mass measurement of the scalar resonance seen at the LHC alone is the mostimportant constraint. In this scenario, the lightest Higgs boson is SM-like and its propertiesare within experimental accuracy. The light singlet scenario is studied in the next section andnon-trivial parameter regions disfavored by Higgs observables and, hence HiggsSignals,are given.

4.4. PHENOMENOLOGY OF A LIGHT SINGLET SCENARIO

In section 3.4 it was pointed out that the MRSSM contains a singlet Higgs boson, whichcould be lighter than the SM-like Higgs boson and several benchmark points were defined

49

4. SUSY Higgs physics

highlighting interesting parameter regions. In this section, consequences for the Higgsphenomenology of a light singlet are discussed. When numerical results are presented,they include all the higher-order effects up to two loops as discussed before.

4.4.1. MIXING OF STATES

To understand the impact on the phenomenology of a lighter than the SM-like Higgs singletstate, the limit of large tanβ and heavy mA is taken. Then, the SM-like Higgs boson isdominantly given by the up-type field φu, and it is sufficient to concentrate on the two bytwo sub-matrix of the Higgs scalar mass-square matrix corresponding to the (φu,φS) fieldsonly, which is

Mφu,S =

m2Z + ∆m2

rad vu

(√2λuµ

eff,−u + g1MD

B

)vu

(√2λuµ

eff,−u + g1MD

B

)m2

S + 4(MDB )2 + λ2

uv2u

2

, (4.9)

where the dominant radiative correction ∆m2rad to the diagonal element of the doublet field

φu, discussed in the previous sections, is included.In the light singlet scenario with mH1 < mH2 ≈ 125 GeV, several consequences may be

derived for the most important parameters appearing in the mass matrix:

Singlet soft mass m2S: The soft breaking parameter m2

S appears in the diagonal elementfor the singlet field φS in the mass matrix (4.9). To ensure that the lightest scalarstate is singlet-like a direct upper limit can be put onto this parameter:

m2S < m2

Z + ∆m2rad , (4.10)

therefore the numerical value of m2S has to be limited to below (120 GeV)2.

Dirac bino-singlino mass MDB : The Dirac mass parameter MD

B is present in the scalarmass matrix due to the supersymmetry breaking term in eq. (2.53). It appears in thesinglet diagonal and in the off-diagonal elements. Hence, a direct upper limit can bederived

(MDB )2 <

m2Z + ∆m2

rad

4≈ (60 GeV)2 , (4.11)

similar to the one on m2S to keep the singlet-diagonal element smaller than the doublet

one. With this conclusion, also constraints on dark matter and direct LHC searchescan be drawn. After all, the bino-singlino mass parameter appears in the neutralinomass matrix and is the dominant parameter that determines the LSP mass in thelight singlet/bino-singlino scenario.

Higgsino mass µu: The higgsino mass parameter µu appears in the off-diagonal elementof the mass matrix in eq. (4.9) in combination with λu. Moreover, µu enters as aparameter in the chargino/neutralino sector, and it is strongly limited from below bythe direct chargino searches and dark matter constraints as is discussed in chapter 6.

Yukawa-like parameter λu: The Yukawa-like parameter λu multiplies the higgsino massparameter µu in the off-diagonal element of eq. (4.9). The off-diagonal element mustbe small enough in order to avoid a negative determinant of the mass matrix and

50

4.4. Phenomenology of a light singlet scenario

hence tachyonic states. In the simple limit vS, mZ µu, avoiding tachyons leads tothe following bound on λu:∣∣∣∣∣λu +

g1MDB√

2µu

∣∣∣∣∣ <mZ

√4(MD

B )2 + m2S√

2µuvu. (4.12)

There might be an accidental cancellation between λu and the g1 term. But it hasbeen shown that dark matter constraints in chapter 6 require µu to be larger than theother dimensionful parameters mZ , mS and MD

B . Hence, the value of λu is necessarilyvery small.

In summary, these simple considerations lead to the following promising parameter hierar-chies for a light singlet scenario:

mS, MDB < mZ < µu , |λu| 1 . (4.13)

Taking these hierarchies into account, the tree-level and one-loop approximations,eqs. (4.3) and (4.7), need to be adapted to be valid in this scenario. The tree-level ap-proximation for the mass of the SM-like Higgs boson H2 in the light singlet scenarioundergoes not only a sign-flip for the singlet contribution but also the denominator needsto be replaced by the difference of the diagonal elements of the mass matrix (4.9). Then,with the same approximations as for eq. (4.3), the formula reads

m2H2,approx, light singlet = m2

Z +∆m2rad+v2

( (g1MD

B

)2

m2Z + ∆m2

rad − m2S − 4(MD

B )2−

(g2MD

W + Λµ)2

4(MDW )2 + m2

T

).

(4.14)For completeness, the approximate mass of the lightest state, which is singlet-like, is thengiven as

m2H1,approx, light singlet = m2

S + 4(MDB )2 − v2

( (g1MD

B

)2

m2Z + ∆m2

rad − m2S − 4(MD

B )2

). (4.15)

The radiative corrections to the doublet diagonal element of the mass matrix ∆m2rad are

assumed to be the dominant ones for the following discussion. At the one-loop level, thecorrection from the top-Yukawa (4.8) is of course the same but the corrections from thenew Yukawa-like superpotential parameters change, as λu needs to be small as does theratio of the singlet and singlino mass parameters appearing in the logarithms, which isclear from the hierarchy (4.13). Hence, only the triplet parameters contribute substantiallyto the one-loop correction and using similar approximations as for eq. (4.7) the analyticalexpression becomes

∆m2H2,eff.pot, light singlet,λ =

2v2

16π2

[5Λ4

8log

m2Ru

(MDW )2

+5Λ4

8log

m2T

(MDW )2

]. (4.16)

The contribution for the top and stops in eq. (4.8) is the same as in the normal orderedscenario.

51

4. SUSY Higgs physics

0 20 40 60 80 100 120

mH1 [GeV]

0.0

0.1

0.2

0.3

0.4

0.5

Z2 H

,12 (

dou

ble

t co

nte

nt)

allowed points

HB excluded

HS excluded

both excluded

Figure 4.7. Shown is the squared mixing matrix element (ZH12)2 of the Higgs mass ma-

trix (C.6) depending on the lightest Higgs boson mass for a scan over parameterpoints. All points of the following fig 4.8 are included. The marker and itscolor show if it is allowed (yellow circle), excluded by HiggsBounds (blue down-triangles), by HiggsSignals (blue up-triangles) or by both programs (violetdiamonds). The upper left of the plot is not populated by points as the SM-likeHiggs boson mass has been fixed to 125 GeV and it is not possible to achievestrong mixing for lower masses of the light state.

4.4.2. HIGGS OBSERVABLES

The doublet-singlet mixing determines how the Higgs boson mass is affected at treelevel by eq. (4.14). The mixing is strongly limited by Higgs observables and searches foradditional Higgs bosons. The study of other Higgs observables besides its mass has beendescribed in section 4.3, and the programs HiggsBounds and HiggsSignals have beenintroduced. These tools are used here to constrain the relevant parameter space of the lightsinglet scenario further. It is obvious that in the state lighter then the SM-like Higgs bosonhas to evade limits from direct searches for light scalars, especially the LEP results [199],requiring a large singlet content. Fig. 4.7 shows if a parameter is allowed (yellow circles)or excluded by HiggsBounds (blue down-triangles), HiggsSignals (green up-triangles) orboth (violett diamonds), depending on the lightest Higgs mass and its doublet content.HiggsBounds restrains the mixing element squared (ZH

12)2 < 0.02 for masses below 80 GeVfrom LEP data. Above this mass the limit becomes significantly weaker as a fluctuationwas measured in the data which weakens the bound. At around 90 GeV, HiggsSignalsconstrains (ZH

12)2 < 0.15 as the SM-like Higgs boson would become too singlet-like abovethat.

Fig. 4.8 shows the resulting excluded and allowed regions in the plane of mS and MDB for

two different values of λu. A variation of Λu is performed to achieve the correct mass for

52

4.4. Phenomenology of a light singlet scenario

0 10 20 30 40 50

MDB [GeV]

0

20

40

60

80

100

mS

[G

eV

]

Allowed (yellow)

Excluded by HB&HS (violet)

Excluded by HB (blue)Excluded by HS (green)

10

20

30

40

50

60

70

80

90

100

110

120

0 10 20 30 40 50

MDB [GeV]

0

20

40

60

80

100

mS

[G

eV

]

Allowed (yellow)

Excluded by HB&HS (violet)

Excluded by HB (blue)Excluded by HS (green)

20

30

40

50

60

7080

90100

110

120

Figure 4.8. The exclusion limits by HiggsBounds and HiggsSignals are given in themS-MD

B parameter plane. For each point Λu has been varied to achieve thecorrect SM-like Higgs boson mass. The yellow region is allowed, while theblue (green) region is excluded by HiggsBounds ( HiggsSignals) and the violetregion by both tools. For the left plot λu = 0.0, while it is −0.01 for the rightplot. All other parameters are chosen as for BMP6.

the observed Higgs boson over the whole investigated parameter region; the remainingparameters are fixed to the values of BMP6. In combination, both plots show that the lightsinglet scenario in the MRSSM is feasible for mS < 100 GeV and MD

B < 55 GeV. This is asimilar range as considered before when using simple approximations for the tree-levelscalar mass matrix, with the exemption of the parameter space with large mixing andlevel-crossing. Moreover, it can be learned from the plots that the parameter λu has aconsiderable effect on the allowed regions. A small change from zero to (−0.01) leads to asignificant change in the allowed regions, which is caused by λu appearing in combinationwith the similarly small term ∝ g1MD

B /µu resulting in possible cancellations.The exotic decays of the SM-like Higgs boson into the light singlet-like states and the

neutralino LSP are summarized in Appendix D. The result is that with the requirementof small mixing between the singlet-like and SM-like Higgs boson also the decays intothe additional bosonic states and the LSP are suppressed and not accessible by currentexperiments. Furthermore, it is ensured that the decay of the Z boson into a pair of LSPs, ifkinematically allowed, is also suppressed and not in conflict with LEP measurements.

4.4.3. MASS CONTRIBUTION FROM MIXING

With the study of Higgs observables it is now easy to understand how much the singlet anddoublet are allowed to mix and still be in agreement with experimental results. This makes

53

4. SUSY Higgs physics

it possible to understand how the mixing can raise the Higgs boson mass at tree-levelfollowing eq. (4.3). In fig. 4.9, the Higgs boson masses (top), the relevant mixing matrixelements squared (middle) and mass splitting compared to the normal ordered hierarchyscenario are shown depending on MD

B (left) and mS (right). The solid green line representsthe SM-like Higgs boson, defined as the more doublet-like state, and the dashed blue linethe singlet-like state. For the mixing matrix elements the solid green (dashed blue) lineshows the doublet (singlet) content of the lightest state and singlet (doublet) content inthe second lightest state, respectively. The contributions to the mixing from the other twostates are numerically small. It can be seen that the hierarchy given in eq. (4.13) identifiedfor MD

B and mS is valid. If the Dirac mass parameter MDB is below 60 GeV or the soft SUSY

breaking scalar mass mS below 60 GeV, respectively, the lightest state is singlet-like. Higgsobservables, as included in HiggsBounds and HiggsSignals, put rather strong bounds onthe singlet mix-in to the SM-like Higgs boson and doublet mix-in to additional scalar states.Hence, the value of (ZH

12)H is bound to be less than roughly 0.2 as can be seen in fig. 4.7,although it also depends mildly on the mass range of the other states. This reduces theupper limit on MD

B and mS to 50 GeV and 70 GeV, respectively.The additional contribution to the mass of the SM-like Higgs boson is above ten GeV for

MDB and about five GeV for mS purely from mixing effects if the mass is close to the level

crossing. Including the limits from Higgs observables, the effect is substantially reduced.Then, the gain to the mass of the SM-like Higgs boson from the tree-level mixing with thesinglet state is expected to be around three to four GeV for realistic scenarios reducing thesize of the needed loop corrections.

54

4.4. Phenomenology of a light singlet scenario

0 10 20 30 40 50 60 70 80

MDB [GeV]

20

40

60

80

100

120

140

MH

i [G

eV

]

0 20 40 60 80 100 120 140

mS [GeV]

20

40

60

80

100

120

140

MH

i [G

eV

]

0 10 20 30 40 50 60 70 80

MDB [GeV]

0.0

0.2

0.4

0.6

0.8

1.0

(ZH H

i,j)2

0 20 40 60 80 100 120 140

mS [GeV]

0.0

0.2

0.4

0.6

0.8

1.0

(ZH H

i,j)2

0 10 20 30 40 50 60 70 80

MDB [GeV]

10

5

0

5

10

mH

,SM

-mH

,ref [

GeV

]

0 20 40 60 80 100 120 140

mS [GeV]

5

0

5

mH

,SM

-mH

,ref [

GeV

]

Figure 4.9. Shown are plots for mixing of the singlet and doublet state in the MRSSMdepending on MD

B (left) and m2S (right). The top panels show the masses of

the singlet-like (dashed, blue) and SM-like state (full, green). The second rowgives the mixing matrix elements for the mix-in of the singlet (up-type doublet)into the lightest state as dashed blue (full green) line. In the bottom row thedifference of the SM-like Higgs boson mass to the mass, when MD

B and m2S

are of a magnitude as in the normal ordered scenario, is shown. All non-variedparameters are chosen as for BMP4.

55

4. SUSY Higgs physics

4.5. SUMMARY

In this chapter different avenues to interpret a 125 GeV SM-like Higgs boson in the MRSSMhave been explored. Even as one of the important mechanisms of the MSSM, the loopcorrection from stop mixing, is forbidden by R-symmetry, the general result is that theobserved resonance can be explained within the MRSSM. To achieve this result, newmechanisms are required.

The MRSSM contains additional states and couplings which affect the mass of the Higgsboson via mixing at tree-level and result in novel loop contributions. It has been shown thatif the discovered state is the lightest Higgs boson, the mixing actually leads to a reductionof the mass. This can be compensated via the loop corrections, and the higher-ordercontributions actually allow to arrive at the experimental value. The new superpotentialparameters λd , λu, Λd and Λu play an important part to obtain the correct Higgs bosonmass as is shown in fig. 4.2 and 4.3.

Alternatively, it is possible that the SM-like Higgs boson is not the lightest scalar state,but a lighter state exists. The singlet of the MRSSM is a natural candidate for it. Then, theHiggs boson mass can get a positive contribution from the mixing of the doublet with thesinglet state as shown in fig. 4.9. This puts constraints on the parameters determining thesinglet-like Higgs boson mass, MD

B and m2S.

Furthermore, the parameter space of the MRSSM has been explored in this chapterusing Higgs observables. Fig. 4.6 shows that the exclusion power of these experimentalconstraints is similar to the MSSM for a comparable choice in the parameter space. In thescenario with a light singlet the observables allow only for a small mixing of the singlet anddoublet as can be seen in fig. 4.7. Therefore, the contributions to the Higgs boson massfrom mixing are rather limited.

56

5. ELECTROWEAK PRECISIONOBSERVABLES

Measurement of electroweak observables are among the most precise in experimental par-ticle physics. Examples for such precision tests are anomalous magnetic dipole momentsof the electron [200] and muon [12]. A wealth of data is available from LEP for electroweakprecision measurements at the Z boson pole [201, 202] as well as the mass determinationsof the electroweak gauge bosons. This information can be used to put stringent constraintson any model of new physics, like the MRSSM.

A description of how the MRSSM is matched to the SM at the Z boson pole massscale has been given in section 3.3. The choice of the electroweak input parametersmZ , Gµ and αMS

SM(mZ ) takes the most precisely measured quantities as input but leavesthe determination of the weak mixing angle θW and the W boson pole mass mW as anon-trivial calculation. In this chapter the procedure of calculating mW in the MRSSMwith the matching to the SM chosen as described in section 3.2 is presented. It followsref. [102], but special care is taken to include the triplet vev which contributes to the mW

prediction already at tree level, see eq. (2.59). To achieve a realistic prediction of the Wboson mass certain reduceable higher-order corrections are resummed and the leadingtwo-loop corrections are taken into account.

In the second part of this chapter the oblique electroweak precision parameters S, Tand U are introduced. They are useful to capture the main contributions of new physicsto processes at the electroweak scale and to compare to experiment. Additionally, it isstraightforward to calculate approximate analytic expressions for them and understand theleading effects and their parameter dependence. Finally, the approximate expressions forS, T and U can also be used to derive a prediction for mW . This is than compared to theresult of the full calculation and helps to identify the main sectors contributing to the mass.The results presented in this chapter are based partially on ref. [145].

5.1. ELECTROWEAK MIXING ANGLE AND MASS OF THE W BOSON

In the following, the calculation of the DR weak mixing angle and the on-shell W bosonmass in the framework of section 3.3 are depicted. It is correct on the complete one-looplevel and takes certain resummable higher-order contributions into account following

57

5. Electroweak precision observables

ref. [203]. Furthermore, leading two-loop SM corrections are included. Then, it is possibleto use this prediction of the W boson mass for comparison with the experimental valueto limit the parameter space of the MRSSM. The experimental value for the W bosonmass [98] is

mW = 80.385± 0.015 GeV , (5.1)

which has almost a 2σ deviation from the SM prediction of

mSMW = 80.360± 0.010 GeV . (5.2)

This value was derived by inserting the input parameters used in this work (A.3) into thefitting formula from ref. [204], and the uncertainty includes the unknown higher-ordercontributions and the variation of input parameters in the 1σ region.

5.1.1. FORMAL ONE-LOOP CORRECTIONS

For the rest of this work the sine and cosine of the DR and on-shell weak mixing angle areabbreviated as follows

sW ≡ sin θW =√

1 − c2W , cW ≡ cos θW =

mW

mZ, (5.3)

sW ≡ sin θW =√

1 − c2W , cW ≡ cos θW =

g2√g2

1 + g22

. (5.4)

From this follows in general that cW 6= mW / mZ where mW , mZ are the DR renormal-ized gauge boson masses given by the other DR parameters using the tree-level rela-tions (2.59). In all formulas of this chapter αMS

SM(mZ ) is already replaced using the threshold

corrections (3.18) by αDRMRSSM(mZ ), which is denoted as α. If not otherwise noted the

renormalization scale is set to the Z boson pole mass µ = mZ .

The parameter ∆r collects all higher-order corrections when calculating the weak mixingangle with the input parameter chosen here as

s2W c2

W =πα√

2Gµm2Z (1 − ∆r )ρtree

(5.5)

with ρtree defined below. To calculate the W boson pole mass from this, several intermediatequantities need to be defined to find the expression for ∆r and mW . The first quantity, theρ parameter, takes into account the difference between on-shell and DR weak mixing angleas

ρ =c2

W

c2W

=m2

W

m2Z c2

W

, (5.6)

which in the SM and MSSM is one at tree level and catches certain higher-order correctionsto the gauge boson masses from e.g. the top quark. As noted before, it differs alreadyfrom one at tree level for the MRSSM. Hence, it is necessary to separate the tree-levelfrom the higher-order contributions. The tree-level ρtree parameter is given as

ρtree ≡1

1 − ∆ρtree, ∆ρtree =

4v2T

v2 + 4v2T

, (5.7)

58

5.1. Electroweak mixing angle and mass of the W boson

while the higher-order corrections ∆ρ are defined by the following relation

ρ =1 − <Π

TW W (m2

W )m2

W

1 − <ΠTZZ (m2

Z )m2

Z

ρtree =1 + <ΠT

ZZ (m2Z )

m2Z

1 + <ΠTW W (m2

W )m2

W

ρtree ≡1

1 − ∆ρρtree . (5.8)

Here ΠTV V (p2) denotes the transversal part of the DR renormalized self-energy of a vector

boson V calculated at momentum p. If terms of the order ∆ρ∆ρtree are neglected, ρ maybe approximated as

ρ =1

1 − ∆ρ − ∆ρtree. (5.9)

The parameter ρ becomes an observable in the MRSSM because of the tree-levelcontribution and an additional experimental input, like mW , could be used to fix vT . Inthis thesis ρ is kept unconstrained and the value of mW is a prediction which needs tobe compared to experiment. In comparison to ref. [203] all appearances of ρ in equationsrelated to higher-order effects need to be replaced by the loop part 1/ (1 − ∆ρ). Then, ∆rcan be calculated as follows

∆r = <(

11 − ∆ρ

ΠTW W (0)

m2W

−ΠT

ZZ (m2Z )

m2Z

)+

11 − ∆ρ

δV B . (5.10)

Here, δV B contains the contributions from the vertex and box diagrams to the muon decayas well as the additional wave function renormalization of the external legs. A detailed studyof δV B can be found in [205]. For the connection of the W boson pole mass to the muondecay another relation is needed with a different set of radiative corrections, collected inthe parameter ∆rW :

s2W =

πα√2Gµm2

W (1 − ∆rW ). (5.11)

Using the connection of ∆rW to the previously defined higher-order parameters

∆rW = ∆ρ(1 − ∆r ) + ∆r (5.12)

allows to calculate the W boson pole mass as

m2W =

12

m2Z ρ

[1 +

√1 −

4πα√2Gµm2

Z ρ(1 − ∆rW )

]. (5.13)

All parameters depend on intermediate results like sW so as described in section 3.3 aniterative algorithm is needed to calculate consistent values.

5.1.2. HIGHER-ORDER QCD CORRECTIONS

For a meaningful prediction of the W boson mass also the leading two-loop QCD-mt

contributions have to be included. This is done following ref. [102] by adding contributionsto ∆r and ∆ρ as of equation (C.5) and (C.6) of this paper. As one of the scenarios examinedhere contains another Higgs state lighter than the SM-like Higgs boson, their respectivemixing has to be taken into account. Hence the contributions from the two lightest Higgsbosons are considered, which modifies equation (C.9) of ref. [102] such that the following

59

5. Electroweak precision observables

replacement is used

ρ(2)(

mh

mt

)→(ZH

12

)2

sin2 βρ(2)(

mH1

mt

)+

(ZH

22

)2

sin2 βρ(2)(

mH2

mt

). (5.14)

The assumption that the heavy Higgs bosons can be taken as decoupled is kept. In thisway the two-loop corrections from ref. [102] read

∆r |2−loop =ααs

4π2s2c2

(2.145

m2t

m2Z

+ 0.575 logmt

mZ− 0.224 − 0.144

m2Z

m2t

)

−13

x2t

[(ZH

12

)2

sin2 βρ(2)(

mH1

mt

)+

(ZH

22

)2

sin2 βρ(2)(

mH2

mt

)](1 − ∆r

) 11 − ∆ρ

, (5.15)

∆ρ|2−loop =ααs

4π2s2

(−2.145

m2t

m2W

+ 1.262 logmt

mZ− 2.24 − 0.85

m2Z

m2t

)

+13

x2t

[(ZH

12

)2

sin2 βρ(2)(

mH1

mt

)+

(ZH

22

)2

sin2 βρ(2)(

mH2

mt

)](5.16)

(5.17)

with xt = 3Gµm2t / (8π2

√2). These contributions add the SM corrections of O(ααs) and

O(α2(m2t / m2

W )2) [206].

5.1.3. THEORETICAL UNCERTAINTIES

To compare the experimental value of the W boson mass (5.1) and the MRSSM predictionin a meaningful way the uncertainty of the MRSSM calculation needs to be understood.The two sources of uncertainty are on the one hand the uncertainty bands on the valueof the input variables and on the other hand the missing knowledge of the higher-ordercorrections. The uncertainty coming from the input values amounts to ∆minput

W ≈ 0.005 GeV.The unknown higher-order corrections contain the complete SUSY two-loop correctionsand sub-leading SM two-loop corrections not captured by the contributions described inthe previous section 5.1.2 and all higher corrections. The latter ones are mainly given bythe ones of O(α2), which are expected to be dominating over the other SM and SUSYtwo-loop contributions. The contributions of this order coming from one-loop correctionsare resummed by the use of eq. (5.13) to calculate the W boson mass. They are actuallycompletely known for the on-shell renormalized calculation using the parameter ∆r as givenin ref. [204], tab. 1, and amount to roughly ten percent of the one-loop O(α) correction. Asa conservative estimate, this contribution is assumed to be leading higher-order uncertaintyand when inserted into the mass formula leads to a value of ∆mhigher order

W ≈ 0.009 GeV.Adding the two sources of uncertainty linearly the complete uncertainty of the W bosonmass in this work is

∆mW = ∆minputW + ∆mhigher order

W = 0.014 GeV . (5.18)

This is of similar size as the uncertainty of the experimental value (5.1).

60

5.2. Electroweak precision observables

5.2. ELECTROWEAK PRECISION OBSERVABLES

The presentation of numerical results for the W boson mass are postponed to the end ofthis chapter. First, a study of the electroweak precision observables is described which alsoallow for a qualitative understanding of the mW dependency on the MRSSM parameters.

5.2.1. INTRODUCTION

A way to describe the influence of new heavy states, which are not directly accessible byexperiments, on the electroweak sector of the model is the use of parameters which arevery sensitive to contributions beyond the SM. Several sets of such variables exist, for thiswork the focus is on S, T and U [207–212]. They are defined by the oblique corrections fromthe new sector to electroweak precision observables and are given by by the contributionsto the propagators of the electroweak gauge bosons. These contributions are exemplifiedby the generic Feynman diagrams in fig. 5.1, where the respective contributing fields needto be inserted. The definitions used here follow ref. [98]:

α(mZ )T =ΠT

ZZ (0)

m2Z

−ΠT

W W (0)

m2W

∣∣∣∣∣new physics

, (5.19)

α(mZ )4c2

W s2W

S =ΠT

ZZ (0) − ΠTZZ (m2

Z )

m2Z

+c2

W − s2W

cW sW

ΠTZA(m2

Z )

m2Z

+ΠT

AA(m2Z )

m2Z

∣∣∣∣∣new physics

, (5.20)

α(mZ )4s2

W

(S + U) =ΠT

W W (0) − ΠTW W (m2

W )

m2W

+cW

sW

ΠTZA(m2

Z )

m2Z

+ΠT

AA(m2Z )

m2Z

∣∣∣∣∣new physics

. (5.21)

Here, “new physics” stands for taking the results of the full model and subtracting theSM part with the approximation that the non-SM contributions and interference effectsare sufficiently small. Because of Ward identities the photon self-energy ΠT

AA(q2) andZ boson–photon mixing self-energy ΠT

ZA(q2) vanish for the beyond SM contributions atzero momentum transfer q2 = 0. Therefore, using their derivatives Π′T , they take theapproximate form

ΠTXY (q2)

∣∣∣new physics

= q2Π′TXY

∣∣∣q2=0

, XY ∈ ZA, AA . (5.22)

Similarly, the differences of self energies for S and U can be approximated using therespective derivatives at zero momentum transfer, which neglects terms of O(q4) whenexpanding the self energies in powers of the momentum,

ΠTXX (0) − ΠT

XX (M2X )

M2X

⇒ −Π′TXX

∣∣∣q2=0

, X ∈ Z, W . (5.23)

S

S

V V

S

V V

V

S

V V

f

f

V V

Figure 5.1. Shown are Feynman diagrams depicting generic gauge boson self-energycontributions.

61

5. Electroweak precision observables

The following discussion is done at the one-loop level not taking into account new physicscontributions of higher order. Moreover, the influence of ∆ρtree is assumed to be smalland treated as an effective one-loop effect. Using S, T and U and further assuming thatthe oblique corrections from new physics dominate over the ones of the vertex and boxcontributions to δV B the mass of the W boson can be written as

m2W = m2

W, SM, ref + δtree-level +αm2

Z c2W

(c2W − s2

W )

(−

S2

+ c2W T +

c2W − s2

W

4s2W

U

). (5.24)

Here, the SM part is the SM prediction and has the value given in eq. (5.2). For the MRSSMalso the tree-level contribution δtree-level from the triplet vev (2.59) has to be added.

5.2.2. QUALITATIVE DISCUSSION

In this section, the contributions of the different MRSSM sectors to the parameters S, Tand U and their relative contribution to eq. (5.24) are discussed. A first understanding canbe grasped from the comparison with leading SM contributions. It is well known that, if nottaken as part of the usual SM, effects from the top and bottom quark to the T parameterdominate the radiative corrections to mW . These contributions take the form [209]

T =NC

16πs2W m2

W

F0(m2t , m2

b) , (5.25)

with

F0(x, y ) = x + y +2xyx − y

logyx

(5.26)

and NC the usual QCD color factor. If the bottom quark mass is neglected and the topmass is written in terms of the Yukawa coupling Yt and vev vu this becomes

Tmb=0

=NCY 2

t v2u

32πs2W m2

W

. (5.27)

As can be seen from eq. (5.27), Yukawa parameters can have strong effects on the obliqueparameters if they are of order one. In the MRSSM, the λ and Λ parameters are newYukawa-like couplings and if they are sufficiently large can have similar effects as the topYukawa. This has already been seen in the context of the Higgs boson mass in chapter 4.

In the following, the limit of large tanβ is taken, additionally, the singlet and tripletvevs are assumed to be small, leading to the hierarchy vu vd , vS, vT ≈ 0. This leadsin general to the case of Λd and λd not appearing in the approximation formulas givenbelow. Under the assumption of small mixing between the up- and down-type sector theirdependence can be restored by replacing the set of parameters vu,Λu,λu,µu, m2

Ru with

vd ,Λd ,λd ,µd , m2Rd

.

MSSM-LIKE CONTRIBUTIONS

The relevant superpartners of the MRSSM similarly appearing in the MSSM, the sleptonsand squarks, give the same contributions as in the MSSM without left-right mixing, forT see e.g. ref. [213]. As example the stop-sbottom contributions are given here which,because of the large top Yukawa coupling, dominate all other corrections and for the limit

62

5.2. Electroweak precision observables

of large soft breaking mass parameter m2Q3

take the form

S =m2

t

12πm2Q3

+O

(m4

t

m4Q3

), T =

m4t

16πm2W m2

Q3

+O

(m4

t

m4Q3

), U =

m4t

20πm4Q3

+O

(m6

t

m6Q3

).

(5.28)As their SUSY masses are at the TeV level, contributions from sfermions are sub-leading tothe ones from other sectors.

Already for these contributions a tendency is visible which repeats itself for most of thecorrections from the other sectors. In lowest order, the U parameter depends on the fourthpower of the Yukawa coupling times a vev (as mt = Ytvu/

√2) over a SUSY mass such

that it becomes dimensionless. The S parameter depends quadratically on this parametercombination. This is also true for the T parameter, but it contains also a factor Yukawacoupling times vev over W boson mass squared as is clear from its definition. Hence, likethe U parameter, T depends on the fourth power of the Yukawa coupling in lowest order,but is only suppressed by a second power of the SUSY mass.

As pointed out in the previous section, fields interacting with the Higgs bosons viasuperpotential parameters and receiving contributions to their mass from electroweaksymmetry breaking contribute to the electroweak precision parameters, like top and stopsin eqs. (5.27) and (5.28). As the unique parameters Λ and λ of the MRSSM are of this form,their contributions are investigated in the following, first for the Higgs and R-Higgs bosonsand then for the neutralinos and charginos.

NEW BOSONIC CONTRIBUTIONS

Higgs sector The Higgs sector depends on the λ and Λ parameters via the singlet andtriplet Higgs fields, additionally the Higgs boson masses are also governed by thesoft parameters m2

S,T , which are large for all the benchmark points considered. Theapproximation taken here is the gaugeless limit, g1 = g2 = 0, and setting λu = 0. Thisleads to µu = Bµ = m2

hu= 0 to fulfill the tree-level tadpole equations. Then, S, T and

U are given as

S = 0 ,

T =v2

u

16πs2W m2

W

Λ4uv2

u

12m2T

+ O

(v4

u

m4T

),

U =v4

u

16πm4T

7Λ4u

30+ O

(v6

u

m6T

). (5.29)

As expected, the expression have a qualitatively similar parameter dependence as inthe stop case (5.28), when the top Yukawa coupling is replaced by Λ and the stopsoft breaking mass by the triplet soft mass.

R-Higgs sector The contributions from the R-Higgs bosons have a structure similar to theHiggs boson contributions. For a qualitative study, the general limit vS = vT = vd = 0

63

5. Electroweak precision observables

is taken and the approximations for the oblique parameters are given as

S =v2

u

16π

[g2

2 +(Λ2

u − 2λ2u)]

3(

m2Ru

+ µ2u

) −g2

2

3(

m2Rd

+ µ2d

) + O

(v4

u

m4R

),

T =v2

u

16πs2W m2

W

[g2

2 +(Λ2

u − 2λ2u)]2

v2u

12(

m2Ru

+ µ2u

) +g4

2v2u

12(

m2Rd

+ µ2d

) + O

(v4

u

m4R

),

U =v4

u

16π

[g2

2 +(Λ2

u − 2λ2u)]2

60(

m2Ru

+ µ2u

)2 +g4

2

60(

m2Rd

+ µ2d

)2

+ O

(v6

u

m6R

).

(5.30)

This is completely analogous to the results for the Higgs sector, showing the samedependencies on the superpotential parameters, vev and relevant soft breakingmasses.

All bosonic contributions due to the Λ and λ superpotential couplings have a similar form andcontribute proportionally to the vev times superpotential parameter to the fourth (second)power for T and U (S), while the S and T (U) parameter are suppressed by a second (fourth)power of the SUSY mass parameter. As with the sfermion contributions, it is expectedthat the T parameter receives the largest corrections. Still, all bosonic contributions aresuppressed by a large soft breaking scalar masses in the scenarios considered here.

A comparable dependency is expected for the fermionic contributions, substitutingthe soft breaking scalar mass parameters by the Dirac masses and µ parameters. Asthe masses of the scalar superpartners are taken to be larger than the masses of thefermionic superpartners, it is to be expected that the fermionic contributions, discussed inthe following, are in fact the leading ones.

NEW FERMIONIC CONTRIBUTIONS

The charginos and neutralinos give potentially large corrections to the oblique observablesas the λ and Λ superpotential parameters appear in conjunction with the Dirac massparameters µd , µu, MD

B and MDW . As especially for the four by four neutralino mass matrix

non-trivial mixing effects can appear, several phenomenologically different scenarios areinvestigated to arrive at simple analytical expressions.

Interplay of Λu and Dirac masses For the simplified case λu = g1 = 0 and µu = MDW the

oblique corrections become

S =2v2

u

16π(MDW )2

12g22 + 19g2Λu + 12Λ2

u

15+ O

(v4

u

(MDW )4

),

T =v2

u

16πs2W m2

W

v2u

(MDW )2

13g42 + 2g3

2Λu + 18g22Λ

2u + 2g2Λ

3u + 13Λ4

u

96+ O

(v4

u

(MDW )4

),

U =v4

u

16π(MDW )4

13g42 + 10g3

2Λu + 106g22Λ

2u + 10g2Λ

3u + 13Λ4

u

84+ O

(v6

u

(MDW )6

).

(5.31)

64

5.2. Electroweak precision observables

For the adjoint case of Λu = g2 = 0 and µu = MDB they are given as

S =2v2

u

16π(MDB )2

8g21 + 7

√2g1λu + 8λ2

u

15+ O

(v4

u

(MDB )4

),

T =v2

u

16πs2W m2

W

v2u

(MDB )2

13g41 + 2

√2g3

1λu + 36g21λ

2u + 4

√2g1λ

3u + 52λ4

u

480+ O

(v4

u

(MDB )4

),

U =v4

u

16π(MDB )4

13g41 + 10

√2g3

1λu + 100g21λ

2u + 20

√2g1λ

3u + 52λ4

u

480+ O

(v6

u

(MDB )6

).

(5.32)

Compared to the sfermion case given in eq. (5.28), there is a similar dependency onthe superpotential parameter, such that Yt is replaced by Λu or λu and the soft SUSYbreaking mass mQ3

by MDW or MD

B , respectively. The T and U parameter show thesame quartic dependency on the Yukawa-like couplings and T is only suppressed bythe square of the SUSY breaking mass. As the limits on the soft breaking squarkmasses are far more stringent than any limit on chargino or neutralino masses, it canbe expected that the contribution from the fermionic sector is dominating with the Tparameter giving the major contribution to mW .

Charginos/neutralinos as vector-like fermions A similar conclusion can be drawn in anexample with non-vanishing couplings using the N = 2 SUSY limit of the MRSSM.Here, g1 = −

√2λu =

√2λd and g2 = Λu = Λd . For a vector-like limit, Λu is set to zero

in the neutralino/chargino tree-level mass matrices and the S–B singlet and the Ru–hu

doublet mix just like the vector-like fermions considered in ref. [214]. Keeping themasses µu and MD

B independent, with x = µu/MDB , the oblique parameters become

S =v2

uλ2u

18π(MDB )2

(3(3x2(1 − x) − 1) log x2

(1 − x)5(1 + x)2

−6 − 4x + 3x2 − 4x3 + 15x4 − 4x5

6x(1 − x)4(1 + x)2

)+O

(v4

u

(MDB )4

),

T =v2

u

16πs2W m2

W

v2uλ

4u

(MDB )2

( (x − 1 − 4x2) log x2

(1 − x)5(1 + x)2

−13 − x + 17x2 − 5x3

6(1 − x)4(1 + x)

)+O

(v4

u

(MDB )4

).

(5.33)

The result for U is an order higher in v2u / (MD

B )2, but as the polynomial is rather lengthyand gives no further insight, it is not shown here. In the limit µu → MD

B , or x → 1,the equations reduce to

S =v2

uλ2u

60π(MDB )2

+ O

(v4

u

(MDB )4

),

T =v2

u

16πs2W m2

W

2v2uλ

4u

5(MDB )2

+ O

(v4

u

(MDB )4

),

65

5. Electroweak precision observables

U =2v4

uλ4u

35π(MDB )4

+ O

(v6

u

(MDB )6

). (5.34)

These results can also be derived from eqs. (5.32), if the limit g1 = −√

2λu is taken.Again, the parameters depend similarly on the superpotential parameter, the vev andthe soft breaking mass parameter as in eqs. (5.28) and (5.31).

No Dirac masses In the limit of no SUSY breaking, where MDB,W = µu,d = 0, the behavior

of the electroweak precision observables changes and they are given as

S =1

6π

[2 + log

2λ2u + Λ2

u

2Λ2u

+ logg2

1 + g22

2g22

],

T =v2

u

16πs2W m2

W

[2λ2

u − 5Λ2u

4+Λ2

u(2λ2u + 3Λ2

u)2λ2

u − Λ2u

log2λ2

u + Λ2u

2Λ2u

+ “g1,2-terms”]

,

U =1

6π

[−12λ4

u + 4λ2uΛ

2u − 15Λ4

u

(Λ2u − 2λ2

u)2−

8λ6u + 4λ4

uΛ2u + 22λ2

uΛ4u + 19Λ6

u

(Λ2u − 2λ2

u)3log

2λ2u + Λ2

u

2Λ2u

]+ “g1,2-terms” . (5.35)

As the soft breaking Dirac masses and the µ parameters are set to zero, the vev isthe only dimensionful parameter that could appear in expressions for the obliqueparameters. The electroweak precision observables are dimensionless, hence thevev can not appear in the formulas for the S and U parameter, if the approximationwith the derivatives (5.23) is used. This is comparable to the result for an additionalSU(2)L doublet given in ref. [209]. The T parameter is defined with a vector bosonmass in the denominator and therefore contains the ratio v2/ m2

W . This is analogousto the dependency of the T parameter on the top Yukawa in eq. (5.27); the logarithmoriginates from the splitting of corresponding chargino and neutralino masses, sim-ilarly as in the known top-bottom result (5.25). The Λ/λ parameters appear in thesame power as the top Yukawa coupling, as is expected from their common origin assuperpotential parameters.

To summarize the previous examples: The neutralino and chargino contributions to theS, T and U parameters can be analytically understood by comparison with the known topand stop result from the SM and MSSM. In the case of no Dirac masses appearing in therespective mass matrices, the electroweak symmetry breaking is the only mechanismintroducing parameters with mass dimension. This is comparable to the case of the topquark in the SM, allowing for an easy understanding of the dependencies of the obliqueparameters as described before.

Without additional Dirac mass terms, some of the neutralinos and charginos would havemasses comparable to electroweak gauge boson masses, which is clearly contradictedby experimental studies [98]. Hence it is necessary to introduce Dirac mass terms asin the first two examples. Similarly to the case of the top squarks in eq. (5.28), Diracmass parameters enter the denominator of the expressions for the electroweak precisionobservables, and are balanced by a vev times Yukawa-like coupling in the same powerfor a dimensionless quantity. As the T parameter contains the parameter in the fourthpower and is only suppressed quadratically by the SUSY masses it gives the dominantcontribution to the W boson mass given by eq. (5.24). This is exemplified in fig. 5.2. Here,the S, T and U parameters from the charginos and neutralinos of equations (5.31) are

66

5.3. Summary

2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0

u

10-5

10-4

10-3

10-2

10-1

100

TSU

2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5

u

10-5

10-4

10-3

10-2

10-1

100

TSU

Figure 5.2. The panels show the comparison of the electroweak precision observablesdepending on Λu, calculated using the approximations given by eqs. (5.31) forthe chargino and neutralino contributions. All other parameters are chosen asfor benchmark point 3 (left, normal ordered scenario) or 5 (right, light singletscenario). The plot for BMP5 does not extend beyond 1.5 as tachyonic statesappear in this range.

plotted depending on Λu while all other parameters are set to the values of BMP3 (left plot)and BMP5 (right plot). For both benchmark points and the whole Λu range, the T parameteris the dominant one. Only if gauge couplings become relevant for vanishing Λu, is theS parameter comparable in size. But this region is disfavored by the Higgs boson masswhich requires a magnitude of Λu close to one. The experimental values of the obliqueparameters are given in ref. [98]:

S = −0.03± 0.10 , T = 0.01± 0.12 , U = 0.05± 0.10 . (5.36)

Taking the 1σ uncertainty region for T this means that |Λu| > 1.5 is disfavored by elec-troweak precision observables for fig. 5.2 when the approximations discussed before areused. To achieve a more complete comparison to experiment, eq. (5.24) shows that the Wboson pole mass can be used as observable to also put constraints on the electroweakprecision parameters, in the case of the MRSSM especially on T .

5.3. SUMMARY

In this chapter, the effects of the extended field content and new coupling structure of theMRSSM have been investigated in context of electroweak precision observables and theW boson mass prediction of the model. The interplay of the oblique parameters S, T andU with the W boson mass is summarized in the following.

In all scenarios considered within this thesis, the Dirac masses are smaller than the softSUSY breaking masses. Therefore, it is to be expected that the fermionic contributionsshown in fig. 5.2 are the dominating ones to the W boson mass given by the approximationformula (5.24). Figure 5.3 shows the W boson mass in black calculated with all contributions

67

5. Electroweak precision observables

2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0

u

80.35

80.40

80.45

80.50

80.55

80.60

80.65

80.70

mW

[G

eV

]

Full resultSM+vT

SM+vT +bosonic STU

SM+vT +all STU

2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5

u

80.35

80.40

80.45

80.50

80.55

80.60

80.65

80.70

mW

[G

eV

]

Full resultSM+vT

SM+vT +bosonic STU

SM+vT +all STU

Figure 5.3. The plots show the comparison of the mass of the W boson depending on Λu,calculated using full MRSSM contributions and different approximations for theelectroweak precision parameters: neutralino and chargino sector, eqs. (5.31)and (5.32), Higgs sector, eqs. (5.29), and R-Higgs sector, eqs. (5.30), as wellas the stop contributions of eqs. (5.28), and the tree-level contribution fromthe triplet vev. All other parameters are chosen as for benchmark point 3 (left,normal ordered scenario) or 5 (right, light singlet scenario).

from eq. (5.13). Furthermore, lines are shown using the approximation formula (5.24) andthe different approximations for the S, T and U parameters as described in the caption. Thereference value for the W boson mass in the approximation (5.24) is its SM prediction (5.2).The parameter Λu is varied and all other parameters are set to either the values of BMP3 orBMP5. The dotted line only takes into account the tree-level contribution from vT to theW boson mass in eq. (2.59). Clearly, it does not describe the full result (full line) well andhigher-order SUSY corrections need to be taken into account. The additional contributionsfrom the sfermions, Higgs and R-Higgs bosons (the dotted-dashed line) do not change thedependency on Λu significantly. Only if the contributions from the fermionic sector are alsoadded, as given by the dashed line, the full result is matched rather closely and the shapecan be reproduced. This validates the previously made statement that the contributionsfrom the charginos and neutralinos, shown in fig. 5.2, and especially the T parameter, arethe most significant. The difference of the approximate to the full result arises mainlyfrom the vertex and box corrections to the muon decay not captured by the electroweakprecision observables.

As in case of the T parameter (see fig. 5.2), so is the prediction for mW 1σ and moreaway from its experimentally measured central value if Λu < −1.5 for the other parameterschosen as for the benchmark points. The region of negative Λ is also favored by theprediction for the SM-like Higgs boson mass as was found in chapter 4. As the magnitudeΛ also needs to be close to unity, the combination of both mass observables, mH1 and mW

provides rather stringent bounds on the parameter space. In chapter 8 these bounds arestudied further including the dependency on other parameter of the model like the Diracmasses.

68

6. DARK MATTER IN THE MRSSM

Dark matter (DM) describes the matter in our universe that does interact gravitationally butis transparent to electromagnetic radiation. Experimental evidence points towards darkmatter being the majority of the matter content in the universe [11]. It is not possible toexplain the abundance of dark matter in the context of the SM. Therefore, dark matterprovides strong experimental evidence to extend the SM even as DM has not yet beendetected directly. Information on the fundamental introduction to DM is taken from refs. [98,215]. For more details see the book [216] and the references therein.

In the first section of this chapter the fundamentals of dark matter in the MRSSM aredescribed. Then, the investigation on how the correct dark matter relic density is achievablein the MRSSM is presented. In the second part the limits from direct DM searches areapplied to the MRSSM and the relevant model parameters are identified. Some results ofthis chapter are also published in ref. [146].

6.1. THE MRSSM AND DARK MATTER

Supersymmetric models have an extended field content and can naturally provide can-didates for dark matter. Besides the richer particle spectrum an additional symmetry isrequired so that not all SUSY particles decay into the lighter SM states. With such asymmetry a lightest stable supersymmetric particle (LSP) exists that can not decay andprovides a perfect DM candidate. This particle falls into the category of a weakly interactingmassive particle (WIMP). WIMPs are a class of DM candidates, which interact only bygravitational and weak forces with a mass between one and several hundred GeV. WIMPproduction in the universe leads to the correct relic density and may provide signals incurrent and the next generation of experiments.

The additional symmetry for the MSSM is usually taken to be R-parity, discussed insection 2.4.1. Then, candidates for DM are the lightest neutralino χ0

1 or sneutrino ν1,where the sneutrino as candidate is actually disfavored by WIMP searches due to itssizable coupling to the Z boson. This would lead either to a small relic abundance as theannihilation is too efficient or a large scattering cross section excluded by direct detectionexperiments [217]. Furthermore, if it is light enough, the gravitino, superpartner to thegraviton, could take the role of the LSP. Many studies for DM scenarios have been done inthe MSSM and references to them can be found in the corresponding section of ref. [216].

69

6. Dark matter in the MRSSM

No additional symmetry has to be added in the MRSSM as R-symmetry automaticallygives rise to an LSP. In the following, the possible scenarios and mass hierarchies of theMRSSM leading to different dark matter candidates are discussed.

6.1.1. DARK MATTER SCENARIOS OF THE MRSSM

There are several possibilities for DM candidates in the MRSSM as fields of differentR-charges are present. The R-Higgs bosons with an R-charge of two and all other MSSM-likeSUSY fields with an R-charge of one have to be treated separately and, depending on themass hierarchy, different scenarios arise.

LSP given by SUSY field with R-charge one and mR01

> 2mLSP: This is the MSSM-likescenario as the list of candidates for dark matter with an R-charge of one is similarto the one of the MSSM. The lightest neutralino or lightest sneutrino could providethe correct relic density in the MRSSM and evade detection by direct detectionexperiments. Additionally, the lightest R-Higgs boson is massive enough to decay intoa pair of LSPs, or possibly into other SUSY fields if kinematically allowed. Therefore,there is exactly one DM state in the model, either a neutralino or sneutrino. It has tobe noted that compared to the MSSM the neutralino as LSP is a Dirac DM candidateinstead of a Majorana one. This has consequences e.g. for the annihilation to lightfermions as opposed to the MSSM there is no helicity suppression of the s-wave inthe MRSSM. Therefore the sfermion exchange becomes a very effective channel.

LSP given by SUSY field with R-charge one and mR01

< 2mLSP: If the mass of lightestneutral R-Higgs boson is below the threshold for two-body decay into a pair of LSPit is also stable as there is no channel through which it can decay into SM fieldsalone. Hence, the dark matter sector in this scenario has two components, a lighterMSSM-like LSP and an heavier but stable neutral R-Higgs boson.

R-Higgs boson is lightest SUSY state: This is similar to the scenario before but the hier-archy between the two dark matter fields is inverted. The MSSM-like DM field cannot decay into an R-Higgs boson and SM fields as R-symmetry forbids this. Therefore,in this scenario the R-Higgs boson is the LSP and the MSSM-like state is an additionalheavier stable DM state.

For the light singlet scenario in section 4.4, a constraint on the bino Dirac mass parameterMD

B < 60 GeV was given by eq. (4.11). This leads to the lightest neutralino χ01 as the LSP

in this scenario. The assumption of χ01 being bino-singlino-like and the DM candidate is

also extended beyond the light singlet case within this work. As can be determined fromthe R-Higgs masses in section C.3, this excludes a lighter R-Higgs boson as the higgsinomass parameter µd and µu parameter appear in the expression for the masses. Moreover,the R-Higgs boson masses include the soft breaking scalar parameters m2

Rdand m2

Ru. The

assumption that all soft breaking scalar masses are of a similar mass scale, close to oneTeV, then puts the masses of R-Higgs bosons much higher than twice the LSP mass ofaround a few hundred GeV. Some parameter regions with a lighter R-Higgs boson might bestill possible due to cancellations but those seem fine-tuned. Thus, the focus in this work ison the first scenario discussed above, assuming that the LSP is the lightest neutralino χ0

1.The scenario of a Dirac neutralino as dark matter candidate in the context of R-symmetric

models has been already studied in refs. [57, 58, 218]. For the MRSSM, the study has been

70

6.2. Dark matter relic density

done by Buckley, Hooper, Kumar in ref. [219], BHK. Using the approximation of four-fermioneffective couplings they showed the promising scenario of a light Dirac neutralino state toachieve the correct relic abundance, albeit requiring high masses for squarks and higgsinosforced by direct search limits. In the following sections, the mechanisms needed to achievethe correct dark matter relic density are presented and the affected model parameters arediscussed. Additionally, recent direct detection limits are used to identify further interestingregions of parameter space. As an extension of BHK this is not done by the applicationof effective couplings, but rather a diagrammatic calculation at tree-level is conducted.This approach takes into account the interference of the different processes for the directdetection experiments, which were neglected by BHK, and leads to an extended parameterspace still allowed by these searches.

6.1.2. IMPLEMENTATION

The calculation of the dark matter relic abundance and matrix elements for the directdetection limits is done using the tool MicrOMEGAs-4.1.8 [220–224]. To carry out thecalculations in the MRSSM, a CalcHEP [225, 226] model file is generated with SARAH,which in turn can be read by MicrOMEGAs. The program includes all other necessarycomponents for the calculation of dark matter observables. For each parameter point in theMRSSM all necessary quantities are conveyed to MicrOMEGAs using a SLHA spectrum file.The prediction of the relic density for a parameter point can then be directly compared tothe experimental measurement. For the case of direct detection the matrix elements arepassed to LUXcalc-1.0.1 [227], which calculates the number of signal events and uses itwith the number of expected background events and measured signal events to derive alikelihood. This is used to determine if a parameter point is excluded as discussed also insection 3.3.

6.2. DARK MATTER RELIC DENSITY

In the early universe the temperature was large enough for the DM particles to be inthermal equilibrium. The abundance of dark matter is created when, with the cooling andthe expansion of the universe, their number density is too diluted for frequent interactions.Moreover, the kinetic energy of lighter SM particles is not high enough to produce theheavier dark matter states. Therefore, the DM freezes out and the number density (nχ) percomoving volume becomes constant, with χ denoting a generic WIMP. The value of darkmatter relic density in our universe is extracted very precisely from studies of the cosmicmicrowave background (CMB). The value given by the Planck experiment [98, 228] is

ΩDMh2 = 0.1186± 0.0020 , (6.1)

where h is the Hubble constant in units of 100 km/(s·Mpc). The baryonic density is given as

Ωbh2 = 0.02226± 0.00023 , (6.2)

which leads to the conclusion that not the known baryonic matter but the unknown darkmatter is responsible for 84% of the mass content in the observable universe. The correctprediction of ΩDMh2 is, therefore, an important test of any complete BSM model. The

71

6. Dark matter in the MRSSM

model should not predict an abundance larger than the measurement, a lower value couldbe allowed if additional sources of dark matter exist.

The determination of ΩDMh2 by MicrOMEGAs relies on the well-known basics of thestandard model of cosmology, thermodynamics in the early universe and the integration ofthe Boltzmann equation. Details on the relevant quantities and calculations can be found inrefs. [215, 216].

The Boltzmann equation in the most commonly used form is given as

dnχdt

+ 3Hnχ = −〈σannv〉(n2χ − n2

χ,eq) , (6.3)

where nχ is the WIMP number density as before, t the time, H the Hubble constant, andnχ,eq and 〈σannv〉 the WIMP equilibrium number density and thermal averaged annihilationcross section, respectively. The second term on the left side takes into account the expan-sion of the universe, while the terms on the right side contain the effects of annihilationsand production.

The model-specific part is the calculation of the cross section that enters the thermalaverage in the Boltzmann equation (6.3):

〈σannv〉 =

∫∞0 dp p24

√p4 − m4

χσ(s)K1(√

s/ T )

m4χT [K2(mχ/ T )]2

. (6.4)

Here, σ is the annihilation cross section, s the usual Mandelstam variable, T the tem-perature, which is used as independent variable instead of the time in the Boltzmannequation (6.3). K1 and K2 are modified Bessel functions.

Besides the annihilation to WIMPs, the process of coannihilation [229–231] can alsobecome important. For this, additional states with masses close to the WIMP mass haveto exist, as it is possible in a SUSY model. Then, scattering of the WIMP with particles inthe thermal plasma can lead to the production of these slightly heavier states. Alternatively,the DM candidate may annihilate with the heavier state. A standard example in the MSSMis the coannihilation of a neutralino LSP χ0

1 with a chargino χ± through the followingprocesses:

χ01τ

− → χ±1 ντ , χ±1 → χ02du , χ0

2χ01 →W +W − . (6.5)

These processes lead to an additional reduction of the relic density of χ01. Further details

on coannihilation scenarios can be found in refs. [216, 232, 233]. For the MRSSM, similarcoannihilation contributions might arise. Modifications like the Dirac nature of the neutralinoand the extended chargino sector have to be taken into account as well as R-chargeconservation when going from this exemplary description to the application for the MRSSM.

6.2.1. PROCESSES IN THE MRSSM

As was stated in the introduction of this chapter, the MRSSM candidate for dark matterconsidered in this work is the lightest neutralino χ0

1 and, motivated by the light singletscenario, it is assumed to be bino-singlino-like. BHK studied this DM candidate withsimplified model assumptions like the N = 2 SUSY limit. They identified the annihilation toa lepton pair through the exchange of sleptons as the most promising channel to achievethe correct relic density. For this MRSSM scenario they show that the mass of the LSP canbe in the range of 10 to 380 GeV taking into account the experimental lower limits on the

72

6.2. Dark matter relic density

Initial state Final state

χ0χ0 `+`−, ν`ν`˜ ˜* γγ, γZν`ν*

` W +W −, ZZ

χ0 ˜* γ`+, Z`+

χ0ν* W +`−, Z ν`

Table 6.1. Initial and respective final states for the coannihilation of the bino-singlino withthe sleptons. For the last two rows the charge conjugate states are also takeninto account. ` ∈ e,µ, τ .

charged slepton masses from LEP [98].

τ

χ01

χ01 τ−

τ+

Z

χ01

χ01

f

f

Figure 6.1. Depicted are Feynman diagrams relevant for the generation of the dark matterrelic density via annihilation through stau or Z boson.

In this work, the result of study by BHK is extended focusing first on the light singletscenario, where the LSP mass is restricted to below 60 GeV. The corresponding Feynmandiagram for the slepton exchange is given on the left of fig. 6.1. The exchange of a Z bosonis identified as another effective channel to achieve the right amount of dark matter in theuniverse, if the LSP mass is around half of the Z boson mass. The second diagram infig. 6.1 shows the Feynman diagram of this process, where the Z boson is created on-shelland decays with the branching factions known from the SM.

In a next step the constraint of mLSP < 60 GeV is dropped and the effect of chargedslepton exchange is studied for the complete relevant mass range to validate the 380 GeVupper limit on the LSP mass presented by BHK.

As mentioned before, coannihilation is another process which can become important forthe DM relic density. This channel was not included by BHK. Relevant coannihilation effectscan be automatically captured by MicrOMEGAs, as described in chapter 16 of ref. [216],and may then be investigated efficiently for the MRSSM. Therefore, it is of interest toinclude them in this work. Two possible scenarios are identified. The first scenario is thecoannihilation of the LSP with a charged slepton with a similar mass. The possible initialand final states for this case can be found in tab. 6.1.

The second scenario assumes that the second-lightest neutralino is wino-triplino-like andclose in mass to the LSP. Then, the coannihilation effects involve also the lightest charginoswhich leads to a complex chain of processes, as has been sketched before for the MSSMin eq. (6.5). As illustration for the MRSSM, the different contributions are given in tab. 6.2.Coannihilation is very efficient in generating the correct DM relic abundance even for highLSP masses, but is applicable only in a very narrow mass range for the next-to-lightestSUSY state.

73

6. Dark matter in the MRSSM

Initial state Final state

χ0χ0 W +W −, `+`−, qq, H1Zχ±χ∓ W +W −, ZZ , ν`ν`, `+`−, qq, q′q′, H1Zρ±ρ∓ W +W −, ZZ , ν`ν`, `+`−, qq, q′q′, H1Z , γγ

χ0χ− q′q, ZW −, H1W −, γW −, `−ν`χ0ρ+ qq′, ZW +, H1W +, γW +, `+ν`χ+ρ+ W +W +

Table 6.2. Initial and respective final states for the coannihilation of the bino-singlino withthe wino-triplino. For the last three rows the charge conjugate states are alsotaken into account. q ∈ u, c, t, q′ ∈ d, s, b, ` ∈ e,µ, τ .

10 20 30 40 50 60

m 01

[GeV]

100

150

200

250

300

mR

[GeV

]

0.050.12

0.1

2

0.50

0.5

0

1.00

1.00

BM4 stau

50 100 150 200 250 300 350 400

m 01

[GeV]

150

200

250

300

350

400

mR

[GeV

]

m0

1=

mR

0.12

0.50

1.0

0

BM3 staus

Figure 6.2. Shown is the dark matter relic density depending on the LSP and right-handedstau mass. The left panel highlights the light singlet scenario, with all constantparameters chosen as for BMP4. The normal ordered scenario is the basis forthe right panel, where all constant parameters are chosen as for BMP3.

6.2.2. RESULTS

The study of dark matter relic density in the context of the MRSSM is summarized infigs. 6.2 to 6.4. The plots in figs. 6.2 and 6.3 show the dark matter relic density as functionof the LSP mass mχ0

1and the right-handed stau mass mτR for both the light singlet and the

normal ordered scenario. The scenario of coannihilation is studied in fig. 6.4, where therelic density is shown in the parameter plane of the LSP mass mχ0

1and the mass splitting

mχ±1− mχ0

1to the lightest chargino, which is assumed to be wino-triplino-like. The left

plot in fig. 6.2 shows the complete result for the light singlet scenario in the parameterplane of the LSP and the right-handed stau mass. An upper limit on the bino-singlino massparameter and therefore the LSP mass of mχ0

1< 60 GeV was derived in chapter 4. The

annihilation modes of slepton and Z boson exchange are of importance in this mass range.The latter mode is relevant if the LSP mass is close to half the Z boson mass, for all othermasses the slepton mode is required for the correct relic density. If the lower mass limit

74

6.2. Dark matter relic density

50 100 150 200 250 300 350 400

m 01

[GeV]

150

200

250

300

350

400

mR

[GeV

]

m0

1=

mR

0.05

0.12

0.5

01.0

0

BM3 no-coann

50 100 150 200 250 300 350 400

m 01

[GeV]

150

200

250

300

350

400

mR

[GeV

]

m0

1=

mR

0.05

0.12

0.5

01.0

0

BM3 sleptons

Figure 6.3. Shown is the dark matter relic density depending on the LSP and right-handedstau mass. All other slepton masses are set to the varied stau mass, theremainder of the parameters are chosen as in BMP3. The left panel does notcontain coannihilation effects, the right panel does.

on the stau mass of 82 GeV [98] is taken into account it can be concluded that the LSPmass needs to be above 20 GeV. Except for the Z mass window discussed before, there isan upper limit on the stau mass in the light singlet scenario depending on the exact LSPmass which does not exceed 150 GeV.

On the right panel of fig. 6.2 the extension of this beyond the light singlet scenario isshown, where BMP3 is taken as basis for the other parameters. If staus are the onlysleptons contributing to the annihilation, there is an upper limit on the LSP mass of 180 GeVbefore the influence on the dark matter relic density becomes to weak to achieve thecorrect value. This also puts an upper bound on the stau mass of at most 200 GeV,depending on the exact LSP mass.

In fig. 6.3 it is depicted how the relic density changes if all sleptons are of similar massand contribute to the annihilation. All sleptons are set to a common mass, which is variedagainst the LSP mass. On the left panel, coannihilation of the LSP and sleptons is notincluded and an upper limit on the LSP mass of 350 GeV for a similar upper limit on thecommon slepton mass is reached. This is close to the LSP mass limit of 380 GeV given byBHK. The difference arises as BHK used a simplified approach while MicrOMEGAs includesmore effects also with regards to dark matter modeling and solving of the Boltzmannequation.

The right panel highlights that this limit does not hold if coannihilation effects are takeninto account. In the case of sleptons with masses close to the LSP mass the relic density isactually enhanced by coannihilation effects. This seem like an unusual behavior as normallycoannihilation reduces the relic density, but it is not unknown to appear in the MSSM [230,232]. Including coannihilation reduces the upper bound on the LSP mass to 300 GeV andan upper limit of 330 GeV for the common slepton mass. This is a non-trivial extension ofthe BHK result.

Even as slepton coannihilation diminishes the upper bound on the LSP mass, fig. 6.4shows that coannihilation with a wino-triplino leads to a welcome reduction of the relic

75

6. Dark matter in the MRSSM

360 380 400 420 440 460 480 500

m 01

[GeV]

10

20

30

40

50

m0 2-m

0 1[G

eV]

0.05

0.12

0.50

BM2 co-annihilation

Figure 6.4. Shown is the dark matter relic density depending on the LSP mass and themass splitting between the LSP and the lightest chargino. All other parametersare chosen as for BMP2.

density to the experimentally allowed region, even for slepton masses in the TeV range.The region where this is possible is limited by lower bounds on the chargino masses fromthe LHC. It is beyond the scope of this work to investigate if there is an upper limit on theLSP mass to achieve the correct relic density with this mechanism. Using the higgsinostates as coannihilation partners might also be possible. But as is shown below, directdetection experiments put severe lower limits on the higgsino mass parameter and mixingwith the bino-singlino, hence this scenario is not pursued here.

6.3. DIRECT DETECTION LIMITS

The sensitivity of DM direct detection experiments to the LSP candidate of the MRSSM,the Dirac-type bino-singlino, is investigated in this section. First, a short overview on theexperimental direct detection observation is given. Then, the relevant processes in theMRSSM contributing to the spin-independent cross section are described.

6.3.1. INTRODUCTION

The details for direct DM searches are comprehensively given in chapter 17 of ref. [216]and are summarized here shortly. If WIMPs are the dark matter of the universe, they aregravitationally bounded in a DM halo around our galaxy. For the astrophysical observationsit is possible to derive the flux of WIMPs on Earth. It strongly depends on the local DMdensity ρ0 at the position of the sun, traditionally taken to be ρ0 = 0.3GeV cm−3 However,the assumptions of the DM halo and the model of the Milky Way used can influencethis value. This can lead to a variation by a factor of the order of two in the density. Thevelocity of DM particles in the halo is conventionally assumed to have a Gaussian velocitydistribution in the standard halo model. For the detection on Earth the velocity has to bebrought into the detector rest frame taking into account Earth’s movement in the Galactic

76

6.3. Direct detection limits

Figure 6.5. Shown is an overview of the direct detection exclusion limits from differentcollaborations. Taken from ref. [234].

rest frame. Due to the rotation of the Earth and the orbit around the sun this can lead to atime and directional dependency of a possible DM signal.

The elastic scattering of WIMPs on nucleons is used for dark matter detection on Earth.This could give a measurable rate for low background detectors even if the interaction ismediated only by the weak force. Figure 6.5 gives an overview on current dark matterdirect detection limits. The result by the LUX collaboration* (green line) sets the moststringent limit on the WIMP-nucleon cross section and also disfavors possible detectionclaims by the DAMA and CDMS experiments, given by the red and orange filled areas,respectively. Therefore, the result of the LUX experiment [235] is used in this thesis todetermine bounds on the dark matter properties from direct detection. It is consistent withthe background-only hypothesis and allows to set limit in the MRSSM parameter space.

LUX is a dark matter detection experiment using liquid xenon as scintillation and ion-ization material. When a WIMP interacts in the detector ionization electrons and directscintillation photons are created. The photons are collected in photomultiplier tubes, while,using an electric field, the electrons drift to a volume of gaseous xenon. There, a so-called“proportional scintillation” signal is produced, delayed from the primary scintillation by theelectron drift time. This approach allows for spatial 3D resolution and signal to backgroundseparation from the ratio of both scintillation rates. The experiment is situated under-ground in the Sanford Underground Laboratory of the Homestake Mine to cut-down onbackgrounds induced by cosmogenic radiation. The liquid xenon tank is surrounded bya water shield for further background suppression from external neutrons. The technicaldetails are also given in chapter 21 of ref. [216].

The input from particle physics to the event rate is given by the differential cross sectionof the WIMPs with the nuclei dσW,N

d ER, where ER is the recoil energy. The scattering cross

section with the quarks and gluons in a nucleon is calculated in the usual quantum fieldtheoretical way. Using hadronic matrix elements this is then transformed to the scattering

*The most recent LUX (2016) result appeared while finalizing this thesis and the most recent limit on thecross section is a factor of five lower than the 2013 result.

77

6. Dark matter in the MRSSM

cross section with nucleons. The WIMP-nucleus cross section gets contributions fromspin-independent and a spin-dependent interactions

dσW,N

d ER=(

dσW,N

d ER

)S.I.

+(

dσW,N

d ER

)S.D.

. (6.6)

The following description is focused on the spin-independent cross section as for theMRSSM it is the most relevant one. For the calculation with MicrOMEGAs also the spin-de-pendent part is taken into account.

The velocity of the WIMPs relative to the nucleons is non-relativistic, and this limit canbe also taken in eq. (6.6). Then, the cross section at zero momentum transfer σDM-N isused to describe the differential cross section as

dσW,N

d ER=

mN

2µ2Nv2

σDM-NF2 (ER)

(6.7)

with the nucleon mass mN , the WIMP mass mχ and the reduced mass µN = mχmN / (mχ +mN ). The nuclear form factor for the coherent scattering F2 contains the information onthe momentum transfer and is related to the nucleon density by Fourier transformation.The spin-independent DM–nucleus cross section at zero momentum transfer is commonlywritten in terms of amplitudes for proton fp and neutron fn:

σDM-N =4µ2

π

(Zfp + (A − Z )fn

)2 , (6.8)

where A and Z are the atomic mass number and atomic number of the nucleon, respectively.In general, these amplitudes contain the information on the contributions of quarks andgluons to the proton and neutron. It was noted by BHK that the Dirac neutralino interactswith the quarks predominantly through squark and Z boson exchange. They show that bothof these interactions can be brought into the form of a vector-vector interaction. As seaquarks and gluons do not contribute to the vector current, the amplitudes fp and fn onlyreceive coherent contributions from the valence quarks [236]. Therefore, the vector-vectorscattering gives the dominate contributions to the direct detection cross section in theMRSSM.

6.3.2. PROCESSES IN THE MRSSM

The expressions of the amplitudes for Z boson and squark exchange are described in thefollowing for the tree-level calculation. The relevant Feynman diagrams are given in fig. 6.6.They are the Z boson t–channel and the squark s– and u–channel exchange.

Z

u/ d

χ01 χ0

1

u/ du/ d

u/ d

χ0,*1 χ0,*

1

u/ du/ d

u/ d

χ01 χ0

1

u/ d

Figure 6.6. Shown are the Feynman diagrams relevant for dark matter direct detectionexperiments.

78

6.3. Direct detection limits

Z BOSON CONTRIBUTIONS

The amplitudes for the Z-mediated diagram on the left of fig. 6.6 are

f Z bosonp =

−g22

8m2Z c2

W

(12

− 2s2W

)(Z1,3−4 + Z2,3−4

2

), (6.9)

f Z bosonn =

g22

16m2Z c2

W

(Z1,3−4 + Z2,3−4

2

), (6.10)

where Zi ,3−4 = (N i13)2 − (N i

14)2 is the difference of the mixing matrix elements squared forthe mixing of the bino with the down- and up-R-higgsino (singlino with the down- andup-higgsino) where i = 1 (i = 2), which can be taken from eqs. (C.26) in the appendix.As before, the notation is sW = sin θW , cW = cos θW . The mixing matrix elements Zi ,3−4

appear in the result as the Z boson couples only to the (R-)higgsino content of the LSP.†

Restricting the (R-)higgsino content of the LSP is required to reduce the magnitude ofthe amplitudes and ensure agreement with current direct detection bounds. This leadsparticularly to constraints on the higgsino mass parameters µu and µd . The direct mixingbetween up-higgsino and bino is dominant and the relevant mixing matrix elements can beapproximated as

Z1,3−4 + Z2,3−4 = −(

g1v2µu

)2

. (6.11)

It clearly demonstrates the suppression by large µu.

SQUARK CONTRIBUTIONS

When the squark-mediated diagrams of fig. 6.6 for general masses of the four first-gen-eration squarks, uL, uR, dL, dR are evaluated it leads to the following expressions for theamplitudes

f squarkp =

g21

4

(1

36((mχ1 + mN )2 − m2uL

)−

49((mχ1 − mN )2 − m2

uR)

)+

g21

8

(1

36((mχ1 + mN )2 − m2dL

)−

19((mχ1 − mN )2 − m2

dR)

), (6.12)

f squarkn =

g21

8

(1

36((mχ1 + mN )2 − m2uL

)−

49((mχ1 − mN )2 − m2

uR)

)+

g21

4

(1

36((mχ1 + mN )2 − m2dL

)−

19((mχ1 − mN )2 − m2

dR)

). (6.13)

For the qualitative discussion the limit of very heavy squarks of equal mass mq = md ≈mu mχ001

, mN is used and the amplitudes reduce to

f heavy squarkp =

g21

4m2q

[(49

−136

)+

12

(19

−1

36

)], (6.14)

†The expressions for the direct detection Z boson exchange derived for equations (6.9) and (6.10) differ by afactor 1/16 from the equivalent expressions derived by BHK.

79

6. Dark matter in the MRSSM

f heavy squarkn =

g21

4m2q

[12

(49

−1

36

)+(

19

−1

36

)]. (6.15)

The minus sign is a product of the different Dirac structure of the s- and u- channels thatcontribute to the process. To achieve agreement with the experimental bounds theseamplitudes need to be suppressed by large squark masses.‡

COMBINATION

To achieve a qualitative understanding of the result, the approximations for the squark ex-change in the limit of heavy squark masses, (6.14) and (6.15), and the Z boson contributions,(6.9) and (6.10), in the limit (6.11) can be added to simple expressions:

f Z boson+heavy squarkn =

g21

96

(7

m2q

−3µ2

u

), f Z boson+heavy squark

p =g2

1

96

(11m2

q

), (6.16)

where s2W = 1/ 4 has been taken in the proton case. Using these results for the dark

matter–nucleus cross section, eq. (6.8), complete destructive interference, σDM-N = 0, canbe achieved if

mq =

√7 + 11A−Z

Z

3µu

Xe≈ 2.2µu , (6.17)

where the numerical value for xenon is calculated with Z = 54 and A = 131.3.In the following, the full numerical calculation is used to specify this sufficient heavy

squark masses and higgsino mass parameter µd and µu and the consequences fromthe destructive interference (6.17) for the direct detection bounds in the MRSSM arepresented.

6.3.3. RESULTS

For the numerical results, MicrOMEGAs is used to calculate the amplitudes for the spin-de-pendent and independent amplitudes taking into account the full model information. Theseare passed to LUXcalc, which derives a likelihood for the direct detection exclusion andtakes into account all astrophysical input as specified in the manual [227].

The results of the direct detection studies are summarized in fig. 6.7. It shows for twodifferent benchmark points (BMP6 and BMP3) with different LSP masses (30 and 250 GeV)the direct detection exclusion bounds in the µu-msquark parameter plane. Furthermore, aline is plotted where the total destructive interference of eq. (6.17) between the Z bosonand squark exchange is attained. The left panel exemplifies the normal ordered scenariowhile the right plot shows the light singlet scenario.

In both plots a funnel shaped region along the line of complete total interference can notexcluded. For the light singlet scenario, given by the left panel of fig. 6.7, µu is stronglycorrelated to the squark masses and, as the LHC puts significant bounds on squarks,§

µu is forced to be larger than 600 GeV by the dark matter constraints. Beyond the light

‡The relative sign between the left- and right-added contributions in equations (6.14) and (6.15) does notappear in BHK.

§The corresponding analysis has not been completed for the case of the MRSSM, but the limits for theMSSM can be used as an conservative estimate[237], putting the limit on the squark mass at 1.3 TeV fordecoupled gluinos.

80

6.4. Summary

400 500 600 700 800 900 1000

u [GeV]

1500

2000

2500

3000

msq

1,2 [

GeV

]

0.050.0

50.

10

0.10

BM3

400 500 600 700 800 900 1000

u [GeV]

1500

2000

2500

3000

msq

1,2 [

GeV

]

0.05

0.05

0.10

0.10

BM6

Figure 6.7. Shown are the 90% and 95% C.L. direct detection exclusion contours (markedby the corresponding p-value) in the µu-msquark parameter plane. The blackline marks the of approximate complete destructive interference between thedifferent processes (6.17) as described in the text. The left (right) plot exempli-fies the normal ordered (light singlet) scenario as all unvaried parameters arechosen as for BMP3 (BMP6).

singlet scenario, the allowed region opens as the cross section limit by direct detectionexperiments is reduced for higher LSP masses as shown on the right of fig. 6.7. There is asimilar bound on µu but the correlation with the squark masses is reduced. The limit onthe squark masses is competitive with the LHC limit for µu > 800 GeV. The difference inthe quantitative bounds becomes clear when using the information from fig. 6.5. There itis shown how the limit on the DM–nucleon cross section is weakened from the minimumin the range of 20 to 60 GeV (BMP6) when going to higher WIMP masses (BMP3). Thisshape originates from the reduction of the dark matter number density with higher massesas the mass density is fixed by the DM halo model.

6.4. SUMMARY

The MRSSM has been confronted with the dark matter evidence and direct detectiondark matter limits in this chapter. The qualitative behavior was studied before by BHKin a simplified approach and has been studied further here with the help of advancedcomputational tools. As an extension to the result of BHK it has becomes clear that theinclusion of more complete scenarios is required and more extensive conclusions can bedrawn.

For the relic density, the upper limit for LSP mass of 380 GeV of BHK could be roughlyreproduced, but only if the coannihilation via charged sleptons is not included. Moreover,the feature of the annihilation via Z boson which loosens the constraint on the stau massesfor a small LSP mass window 42 GeV < mχ0

1< 48 GeV. It has been shown that the

inclusion of coannihilation effects is important as coannihilation with sleptons actuallyenhances the relic density. This leads to a reduction for the LSP mass upper bound to

81

6. Dark matter in the MRSSM

about 300 GeV. Instead, the coannihilation with almost mass-degenerated wino-triplinostates provides a mechanism give the right dark matter relic density with an LSP massbeyond 380 GeV.

The direct detection rate of the Dirac-type LSP in the MRSSM is driven mainly by Z bosonand squark exchange. As the first channel couples through the higgsino components thisputs a bound on the higgsino mass parameter µu, which is relevant for tanβ 1. Onthe other hand, the squark channel directly puts a limit on the masses of first-generationsquarks. This behavior is also given by BHK using an effective field theory approachwith tree-level couplings. For this thesis, the expressions were rederived finding somedifference in prefactors. Additionally, interference effects using the full model informationare included here, which was not done in BHK. These effects are crucial in a funnel shapedregion in the µu-msquark parameter space allowing µu and squark masses of moderate size.

The combination of both dark matter observables constrains the parameter space of theMRSSM considerably as the bino-singlino is the most promising dark matter candidate inthe model. Direct searches limit µu to be above roughly 500 to 600 GeV and the squarkmasses of the first generation to be higher than 1.3 TeV (see fig. 6.7). The dark matter relicdensity limits the LSP mass to be below 300 GeV for annihilation via sleptons. Furthermore,the slepton masses are bound to be below 100 to 280 GeV depending on the LSP mass,see fig. 6.3. If only staus are relevant the annihilation, the LSP mass is limited to be atmost 180 GeV while the stau masses have an upper limit of roughly 200 GeV dependingon the LSP mass, see fig. 6.2.

Coannihilation with almost mass-degenerated wino-triplinos can improve the upperbound on the LSP mass to at least 500 GeV as confirmed with fig. 6.4 and there is noapparent problem with extending this limit to higher LSP masses, but verifying this factgoes beyond the scope of this thesis. The main drawback of this scenario is that thewino-triplino mass parameter is limited to be close to the bino-singlino mass parameter,MD

W ≈MDB . The limit on µu and the mixing of the bino-singlino with the higgsino does not

allow for a higgsino to be an efficient coannihilation candidate.

82

7. DIRECT SUSY SEARCHES AT THELHC

To achieve a deeper understanding of the foundation of nature and find a description thatgoes beyond the SM (BSM) it is necessary to probe energy scales above the electroweakscale. The Large Hadron Collider (LHC) is the largest man-made experimental apparatusand allows for the first time to probe the multi-TeV mass range directly in a controlledenvironment.

In chapters 4 and 5 the MRSSM was probed mainly by relying on measurements andlooking for deviations from SM observables, like the Higgs and W boson masses as wellas electroweak and Z boson peak precision observables. This is an indirect way to lookfor BSM physics which is sensitive only to a certain combination of masses and couplingparameters and not easy to identify which sector leads to a certain effect. Direct searchesfor unknown particles provide a straightforward way to access BSM physics or at leastallow to set limits on the respective masses and the mass scale of the model.

Currently, the LHC is the best place to look for BSM physics. It is a proton colliderring with four major detectors: ATLAS (A Toroidal LHC ApparatuS), CMS (Compact-Muon-Solenoid), ALICE (A Large Ion Collider Experiment) and LHCb (Large Hadron Collider beauty).The LHC and its experiments are situated at CERN, the European organization for nuclearresearch.

In this chapter, results from the ATLAS experiment are interpreted in the context ofthe MRSSM. First, a short introduction of the LHC and possible SUSY signatures is given.Secondly, the phenomenology of the MRSSM at the LHC is discussed. Then, results fromATLAS regarding searches for new heavy electroweak states are interpreted in context ofthe MRSSM. The results of this chapter were first presented in ref. [146].

7.1. SUSY PHENOMENOLOGY AT A PARTICLE COLLIDER

Results from the LHC have already been used in chapter 4 to constrain the parameterspace from Higgs signal strength measurements and via direct searches for additionalstates. In the following, the production of new SUSY states in proton proton collisions isdiscussed. First, the experimental apparatus is introduced and possible supersymmetricsignatures described. The description is based on ref. [238].

83

7. Direct SUSY searches at the LHC

7.1.1. THE LHC AND EXPERIMENT DESCRIPTION

The LHC is a proton storage ring and, as a hadron collider, planned as a discovery machinefor new physics. It has a design center-of-mass energy of 14 TeV and luminosity of1034 cm−2 s−1 which may be reached in the coming years with the on-going Run 2 andbeyond. For this work, results from Run 1 from 2010 to 2012 are used. In that time, at firstan integrated luminosity of 4.6 fb−1 were collected at a center-of-mass energy of 7 TeV andin the last year 20.3 fb−1 integrated luminosity was achieved at 8 TeV.

In this chapter all experimental results are taken from the ATLAS collaboration. The struc-ture of a multi-purpose experiment like the ATLAS detector is described comprehensivelyin the following.

Tracking detector The part of the detector closest to the interaction region is the trackingsystem. Using a magnetic field the tracks of charged particles are bent allowing for amomentum measurement of charged particles.

Calorimeter The next layers are the electromagnetic and hadronic calorimeters. Theirpurpose is to measure energy and direction of electrons, photons and hadrons. Thisis achieved by inducing electromagnetic and hadronic showers in the calorimetersoriginating from these particles. The energy deposition from the showers in thecalorimeters allows to measure the energy and direction of flight of the initial particle.

Muon systems The muon system is an additional layer of tracking detector. It is situatedat the outer zones of the experiment and immersed in a magnetic field to allow formomentum measurement. Muons, as minimally ionizing particles, are not capturedin the calorimeters and transverse the detector. The additional tracking on the outsideof the calorimeters allows for a higher momentum resolution of the muons.

Particle identification and triggering Combining the tracking and calorimeter data allowsfor the identification of final state particles and physics objects like charged leptonsand jets and obtaining information about them like their momentum and charge. Thisdata is recorded for every event and due to the high bunch-crossing rate is far to largeto be stored permanently. A filter is needed so that only the information of interestingevents is kept. This is achieved by several layers of trigger systems which only selectevents with interesting physic. The triggers reduce the event rate from 400 kHz to≈ 200 Hz for writing to permanent storage for further study.

More details on the detector and experiment can be found e.g. in chapter 3 of ref. [238]and the technical design report of the ATLAS experiment [239, 240].

With the data containing the physics objects available in storage the collaborations canstudy them in detail. Searches for BSM physics at collider experiments are usually doneusing a cut-based strategy. Properties which can separate BSM signals from the SMbackground are identified for this approach. Cuts on the relevant physics objects like thetransverse momentum pT or angular variables as well as composite quantities are thenused to remove events that are not signal-like. Monte-Carlo studies, and for the backgroundalso data-driven methods, are employed to estimate the expected number of signal andbackground events, respectively. With these numbers exclusion limits can be calculated asoutlined in section 3.3.

84

7.1. SUSY phenomenology at a particle collider

7.1.2. EXPERIMENTAL SUSY SEARCHES

Supersymmetry can appear in manifold ways at the LHC. One of the factors deciding therelevant event topology is if a lightest stable SUSY particle (LSP) is present in the model ornot. In this section a short review of SUSY searches for models with an LSP is given, asR-symmetry leads to such a lightest stable SUSY state. As motivated by the dark matterstudies in chapter 6, it is assumed in this thesis that the LSP is the lightest neutralino χ0

1,which is mainly bino-singlino-like.

Events with supersymmetric particles at the LHC are the pair production of high massSUSY states. The R-charge of all SM particles is zero. Hence, the initial state quarks andgluons have no R-charge and the produced SUSY states then have opposite R-charges.They subsequently decay into SM particles and a pair of LSPs. The LSP does not decayfurther and only interacts weakly, therefore it escapes the experiment undetected exceptfor the measurement of missing transverse energy Emiss

T comparable to the neutrino in theSM. Neutrinos interact only via the weak force and hence escape the detector undetected.The presence of a neutrino is indicated by Emiss

T , which is the negative of the sum overall the momenta of particle in the event projected onto the transverse plane. Due tomomentum conservation in this plane, this quantity may be matched to the momentum ofthe neutrino or the sum of momenta if several neutrinos are expected in the event.

All search strategies which are relevant and commented on within this work have arequirement of a large Emiss

T in signal events in common. In following, the production anddetection of strongly and electroweakly interacting SUSY particles at the LHC is discussed.

STRONG SECTOR

At an hadron collider, strongly interacting particles are produced predominantly. Hence,squarks and gluinos should appear in large abundance, if their masses are in an accessiblerange. As the LSP is a colorless state the decay chains of squarks and gluinos compriseseveral quarks and gluons. These hadronize and leave signatures as jets in the hadroncalorimeter. The most straight-forward way to find SUSY at the LHC is therefore to look forevents with jets and Emiss

T . Having the largest signal rate, this signature is also plagued bythe largest background as jets can be produced in many ways at a proton-proton collider.The Emiss

T may stem from the irreducible background of a Z boson decaying to neutrinos.Reducible background can come from W boson decays with not-identified leptons or viahadronic tau decay, also the jet energy may be miss-measured and lead to an imbalance inthe transverse plane.

To reduce the especially large pure QCD background additional leptons, which in thecontext of the LHC usually means electrons and muons as they are easier to identify thantaus, can be required in the final state. They are produced from the decay of W or Zbosons if intermediate charginos or neutralinos appear in the decay chains. In analyses forSUSY-QCD events usually up to two leptons are required. Especially for the case of twoleptons with the same charge assignment the SM background is reduced. See refs. [237,241–243] for examples of ATLAS analyses. The production of the scalar octet state, thesgluon, is a special case as it has an R-charge of zero and is discussed below.

85

7. Direct SUSY searches at the LHC

ELECTROWEAK SECTOR

The production of the non-strongly interacting SUSY states is only possible by the quark-quarkinitial state. Therefore, the cross section to produce them is suppressed compared to thesquark and gluino production. Hence, electroweak SUSY processes become important ifthe mass range of the SUSY-QCD sector is out of experimental reach. Then, the productionof sleptons, charginos and neutralinos and their subsequent decays need to be investigated.

The gaugino content of charginos and neutralinos can couple directly to the quarks of theinitial protons while the higgsinos would interact with a negligible Yukawa coupling and areproduced only via Drell–Yan processes. Charginos and neutralinos decay in manifold waysto the LSP. These decay chains contain Z, W or Higgs bosons, depending on the nature ofcharginos and neutralinos as gauginos or higgsinos. They might also include intermediatesleptons if these are light enough.

Additionally, sleptons may be produced directly in the s-channel by intermediate SMgauge bosons but decay into final states containing leptons and the LSP escaping thedetector leading to signatures of Emiss

T . Light staus are especially motivated from darkmatter studies, see e.g. chapter 6, as they help to achieve the correct dark matter relicdensity, both in the MRSSM and the MSSM. With this information final states with a largemultiplicity of tau leptons are expected if a realization of supersymmetry is accessible atthe LHC. Background events to the production of electroweak SUSY states with a signatureof several leptons can be produced by diboson (ZZ, WW or ZW) events, W or Z bosonassociated with jets, as well as the top pair production with leptons from secondary decaysfrom the hadronized final states. See refs. [244] for a summary of ATLAS analyses.

SIMPLIFIED MODELS

The parameter space of a supersymmetric model relevant for searches at the LHC ismultidimensional and it is non-trivial to put general mass limits on the production of certainSUSY states. Hence, it is a common procedure by the experiments to interpret their resultsin full models with limiting constraints applied. For example, the Constrained MSSM(CMSSM) has only five input parameters. The problem with such model-dependent studiesis that due to the constraints they put also very strong indirect limits on all the other sectorseven if direct search limits are almost an order of magnitude weaker. Because of thisthe use of so-called simplified model studies [245] has become common. These studiesconsider the parameters which can be directly inferred from experiment. For illustration,the production of charginos and neutralinos in ref. [246] is studied in the simplified modelscenario where only the lightest chargino χ±1 is produced and decays via a W boson tothe LSP: χ±1 →W±χ0

1. This allows to set clear limits on the masses of the involved statesand has been done for many different combinations of channels by the experiments, seerefs. [237, 244].

The challenge for a phenomenological study then is to connect the results from the simpli-fied model studies to actual realistic models like the MRSSM. This is not straightforward asa SUSY state does normally not decay with a branching ratio of one into a certain final state.The prime example is the decay of a neutralino into either a Z or an Higgs boson and theLSP. In ref. [247] the topologies pp→ χ±1 χ

02 →W±χ0

1 Zχ01 and pp→ χ±1 χ

02 →W±χ0

1 Hχ01

have been interpreted as simplified model studies with degenerate masses for χ±1 andχ0

2. For the first topology an upper limit of roughly 350 GeV can be set on this commonSUSY mass, while in the case of the second topology only a limit of roughly 150 GeV is

86

7.2. The MRSSM at the LHC

expected. This is due to the very different decays of the SM bosons and the abundance ofproduced leptons which are most relevant for the searches. A neutralino might not decaypredominantly in one or the other SM boson but with varying branching ratios dependingon the parameter point of the model leading to a strongly varying upper limit on the massof the state. For a realistic model it is therefore necessary to combine these two resultsin a meaningful way. This complexity shows that it is not possible to derive universalmodel-independent limits on the masses of new physics states from collider searches. Aninterpretation in a certain model is always needed to connect the experimental result totheory.

7.2. THE MRSSM AT THE LHC

After the previous general remarks on supersymmetry at the LHC an overview of thespecific collider signatures of the MRSSM is given in this section. In the second part it isdiscussed how the experimental results published by experiments can be applied to theparameter space of the MRSSM.

7.2.1. MRSSM SIGNATURES

MRSSM states relevant for LHC searches are as in the MSSM the strongly interactingsquarks and gluinos and the electroweakly interacting charginos, neutralinos and sleptons.Additionally, the R-Higgs bosons as well as the CP-even, CP-odd scalar color octets andHiggs bosons with no R-charge may be accessible by the LHC.

STRONG SECTOR

It is known from the MSSM that for processes with strongly interacting squark and gluinoslarge uncertainties are present at tree-level and corrections from higher-order effects needto be considered [248, 249]. For the MRSSM similar corrections are to be expected, butdifferences of the models have to be taken into account like the Dirac fermion nature of thegluino in the MRSSM compared to the Majorana nature in the MSSM. Another differencestems from R-symmetry compared to R-parity. For the production of two squarks ( or twoanti-squarks) all of the three combinations qLqL, qRqR, qLqR are possible final states in theMSSM. In the MRSSM, only the last combination is allowed to be produced as the finalstate has a total R-charge of zero, while it is non-zero in the other two cases. Additionally,the color octets contribute at the quantum level to the processes and the running of thestrong coupling constant αS. These differences lead to an absolute shift in the total crosssection when comparing both models. Also, the contributions from different channels canvary relative to each other. Because the calculation needs to be done at next-to-leadingorder in αS it goes beyond the scope of this thesis. and results will be presented in aforthcoming publication [250].

The scalar color octets might also be the kinematic reach of the LHC. As they have anR-charge of zero, they decay fully in SM particles and the search strategies discussed abovedo not apply as no relevant Emiss

T contributions are generated. The case of color octets atthe LHC is considered independent of the MRSSM in refs. [55, 251–254]. In the case ofthe MRSSM, the scenario has been studied in refs. [56, 205, 255, 256], and limits from thelatest LHC runs and predictions for the future runs have been derived.

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7. Direct SUSY searches at the LHC

ELECTROWEAK SECTOR

The production of the R-Higgs bosons was considered in ref. [61]. R-Higgs bosons mayonly be produced in pairs via a Drell–Yan process as they have an R-charge of two. Ifthey decay further the final state contains four LSP, leading to large Emiss

T in the events.Furthermore, when the staus are considered to be light as required for dark matter limitsshown in chapter 6, a high lepton multiplicity can be expected in the final state of R-Higgsproduction.

The Higgs sector of the MRSSM has been studied in the context of LHC physics inchapter 4 mainly with regards to the SM-like Higgs boson and the possibility of a lightersinglet-like state. It is of course a viable possibility that also other heavier Higgs bosons ofthe MRSSM are accessible by the LHC. This includes MSSM-like CP-even and CP-odd aswell as two charged Higgs bosons. Furthermore, if not lighter than the SM-like Higgs boson,a singlet-like scalar and pseudo-scalar as well as a CP-even, a CP-odd and four triplet-likecharged Higgs bosons may be in LHC reach. The Higgs sector of the MRSSM is morediverse than the one of the MSSM. It can be expected that if there is a discovery, it will notbe a single state but several appearing in many different channels. This would provide avast testing ground for predictions from the MRSSM.* A discovery of more MRSSM Higgsbosons in the high mass region with a higher statistics is not ruled out.

The production of electroweakly interacting charginos and neutralinos with Dirac massterms and sleptons in an R-symmetric theory were discussed in ref. [62] in the contextof linear colliders like the International Linear Collider (ILC). Prospects for the LHC werenot considered there. This is done in detail in section 7.3 using ATLAS results from Run 1,where energies of 7 and 8 TeV were reached. The implementation of how the experimentalresults were interpreted and limits in the MRSSM parameter space derived is given in thefollowing.

7.2.2. IMPLEMENTATION

In recent years different computational tools emerged, which aim to automatize the studyof BSM physics at the LHC in a generic way using standardized interfaces. Two promisingapproaches have evolved that rely on different output from the experimental collaborations.

• The first way uses the efficiency tables publish for different analyses by the ex-periments [261]. These tools combine the results of simplified model studies in aconsistent and weighted way to set limits on the parameter space of new physicsmodels which allows a straightforward and computationally fast recasting of exper-imental results. This approach suffers from the drawback that it is not fully modelindependent, as for the calculation of the efficiencies knowledge of a specific modelof new physics has to be included. Examples for such tools are SModelS [262] andFASTlim [263], which are so far only available for the MSSM. It would take a detailedstudy to apply them to other models like the MRSSM.

• Approaches to exclude assumptions of the BSM model in the implementation go onestep back. The tools rely on the model-independent information: Either the expected

*There were candidates for a possible observation at the LHC in the previous years which proved as statisticalfluctuations. At the end of Run 1, there was an excess at 2 TeV for hadronically decaying diboson states [257,258] which does not appear in the Run 2 data. After the first year of Run 2 a possible resonance was seenat 750 GeV in the diphoton spectrum [259, 260], which vanished in the next year.

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7.2. The MRSSM at the LHC

process χ0χ0 χ0χ− χ0ρ+ χ+χ0 ρ−χ0 χ+χ− ρ+ρ− ˜+R,i

˜−R,j

BMP4 496 619 3.9 1147 0.83 496 1.25 17.6BMP5 4.77 5.58 10.0 15.6 1.99 7.93 2.88 1.48 × 10−3

BMP6 1.67 6.04 22.6 17.0 5.65 12.1 8.48 38.4

Table 7.1. Electroweak cross sections at a center-of-mass energy of√

s = 8 TeV ascalculated by Herwig++ using the SARAH model input. All values given in fb. Thecross sections are calculated at leading order with uncertainties of the order often percent determined from scale variation. The integration error is below onepercent.

limit on the signal cross section or the number of background events and observedevents in certain signal regions can be used and both are provided by experimentsin their studies. Then, it is necessary to derive an estimate for the number ofsignal events including the experimental acceptance and efficiency for a grid ofparameter points. For this step a Monte-Carlo generator and a parametrized detectorsimulation are usually employed. Combining all the numbers allows to set limits inthe explored parameter space. Examples for such programs are CheckMATE [264]and MadAnalysis [265, 266].The downside of this approach is that it requires more computing resources thanthe first one. Also, the used detector simulations like Delphes [267, 268] are notofficially tuned by the experiments to match the detector setup but only aim for anapproximate fit. It still turns out that this is enough to reproduce exclusion boundsset by the experiments in simplified models and therefore should suffice to set limitsin a so far untested model like the MRSSM.

For the study of electroweak MRSSM processes at the LHC the approach described in thelast point is used. It allows to take the model differences between the MRSSM and theMSSM or the simplified models studied by the experiments into account and to directlycalculate bounds from the LHC in the MRSSM parameter space.

Using the UFO [269] output produced by SARAH with Herwig++-2.7 [270, 271] the produc-tion of electroweak SUSY states at the LHC is simulated at leading order. The generatedevents are used with CheckMATE-1.2.0 to compare with 8 TeV data. CheckMATE includesATLAS analyses, which can be sensitive to the final state of processes of interest, if severalleptons appear as decay products. Specifically, for the purpose of this study the analysesimplemented in CheckMATE for two [246, 272], three [247, 273] and four and more leptons[274] in the final state are considered here. The final output of CheckMATE used to setexclusion limits in the MRSSM parameter space is the value for CLS of most sensitivesignal region of the studied analyses for each parameter point.

To ensure correctness of the calculated limits several tests of the used tools weredone. For the calculation of the matrix element the event generator Herwig++-2.7 waschecked against MadGraph5 _ aMC@NLO [275], where the input of model files was alsogenerated with SARAH. Agreement was achieved up to the implementation differences ofthe programs. This can be seen when comparing the numbers in tab. 7.1 and tab. 7.2.

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7. Direct SUSY searches at the LHC

process χ0χ0 χ0χ− χ0ρ+ χ+χ0 ρ−χ0 χ+χ− ρ+ρ− ˜+R,i

˜−R,j

BMP4 495 623 3.72 1142 0.57 490 0.79 21.6BMP5 4.70 5.65 9.06 15.6 2.07 7.73 2.94 1.46 × 10−3

BMP6 1.64 6.05 23.2 16.7 5.50 12.0 8.22 40.7

Table 7.2. Electroweak cross sections at a center-of-mass energy of√

s = 8 TeV ascalculated by MadGraph using the SARAH model input. All values given in fb. Thecross sections are calculated at leading order with uncertainties of the order often percent determined from scale variation. The integration error is below onepercent.

7.3. ELECTROWEAK LHC SEARCHES IN THE MRSSM

A SUSY mass spectrum accessible by Run 1 electroweak searches at the LHC requiresa rather low LSP mass around or below some hundred GeV. In the light singlet scenariothis requirement is fulfilled automatically via eq. (4.11). Therefore, only the light singletscenario is considered in this section.

With the setup discussed in section 7.2.2 the following production processes pp→ ˜+R

˜−R,

pp → χ0χ0, pp → χ0ρ−, ρ+χ0, pp → χ0χ+, χ−χ0 can be studied at the LHC. Fig. 7.1illustrates the production of the electroweak SUSY states of interest from the interactionof valence or sea quarks coming from the protons. In fig. 7.1 the Feynman diagrams forthe two- and three- body decays of the charginos, neutralinos and sleptons are shown.As discussed before, they contribute with different branching fractions depending on thecomposition of the charginos and neutralinos as well as the mass hierarchy of the SUSYstates.

q

q

p

p

χ, ρ, f

χ, ρ, f *

Figure 7.1. Illustration of the production of electroweak SUSY states at the LHC. Theincoming protons provide the necessary quark and anti-quark. These inter-act electroweakly to produce a pair of charginos and neutralinos or a pair ofsleptons.

The corresponding production cross sections of the relevant benchmark points are givenin tab. 7.1. Processes with left-handed sleptons are neglected, assuming that the massesare above the detection limit while right-handed sleptons might be accessible at the LHCdue to the conditions from the dark matter sector shown in chapter 6. The masses of therelevant particles are given in Tab. 7.3.

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7.3. Electroweak LHC searches in the MRSSM

˜±

χ0

`±

χ0

Z0, H0, A0

χ0

χ±, ρ±

W±, H±

χ0

˜χ0

`±

¯±

χ0

˜χ±, ρ±

`±

ν

χ0

Figure 7.2. Relevant Feynman diagrams for the decay of the electroweak SUSY states.

χ01 χ0

2 χ03 χ0

4 χ±1 χ±2 ρ±1 ρ±2 τR µR eR ˜L mH1

BMP4 49.8 132 617 691 131 625 614 713 128 802 802 808 100BMP5 43.9 401 519 589 409 524 519 610 1000 1001 1001 1005 94BMP6 29.7 427 562 579 422 562 433 587 106 353 353 508 95

Table 7.3. Masses of the non-SM particles in the BMPs relevant for the LHC studiesdiscussed here. All values given in GeV.

7.3.1. DISCRIMINATING THE MRSSM FROM THE MSSM

A first step in understanding the MRSSM in context of LHC physics is to gain an insighthow the MRSSM electroweak sector differs from the one of the MSSM. This is studied inthe following by first understanding the differences in the mass eigenstates of the MRSSMand MSSM. In a second step an example scenario is compared between both models.

UNDERSTANDING THE DIFFERENT PHENOMENOLOGIES

Several differences are evident when comparing the chargino and neutralino sectors of theMSSM and the MRSSM. In the MRSSM, the neutralinos are Dirac fermions composedof eight Weyl spinors ξi = (B, W 0, R0

d , R0u) and ζi = (S, T 0, H0

d , H0u ). Assuming that the

SUSY breaking mass parameters (MDB , MD

W ,µd ,µu) are larger than the electroweak scaleMZ they contribute dominantly to the four mass eigenvalues, see eq. (C.26) for the massmatrix. In contrast, in the MSSM the neutralinos are of Majorana type, and there is onlya single higgsino mass parameter µ, so that two neutralino masses are approximatelydegenerate. The MRSSM contains four different charginos, composed of eight Weylspinors (T −, H−

d ), (W +, R+d ), (W −, R−

u ), (T +, H+u ), with masses determined by the wino and the

two higgsino mass parameters, see eqs. (C.28) and (C.31) of the Appendix. In comparisonthere are only two states in the MSSM.

The naive expectation from having doubled the number of degrees of freedom ofneutralinos and charginos compared to the MSSM is an enhancement of the pair productioncross section by roughly a factor of four. This is not true as R-symmetry, unlike R-parity,forbids several final states. So, due to R-charge conservation, only half of all final statecombinations are actually allowed as the initial state R-charge is zero. Moreover, the newgenuine MRSSM states (Rd , Ru, S, T ) do not interact at tree level with fermions, sfermionsor gluons. Hence the situation is more complicated and needs to be analyzed separately

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7. Direct SUSY searches at the LHC

for each channel.Not only the production of charginos and neutralinos differs between MRSSM and

MSSM. The corresponding branching ratios are also distinct in both models as the mixingbehavior and coupling structure varies. As an example, in the MSSM the higgsino massparameter µ induces a maximal mixing between the up- and down-higgsino. Therefore,both the combined states mix almost in the same way with the bino and wino. This givescomparable decay rates into sleptons for both states, if kinematically available, as theycouple mainly to the wino and bino component. In contrast, for the MRSSM R-symmetryforbids the µ parameter which induces mixing between the up- and down-(R-)higgsinos.Instead, up- and down-(R-)higgsino states have mass parameters µu and µd and giveseparately the main contribution to the corresponding mass eigenstates. Furthermore, inthe scenario of tanβ 1 a suppression of the mixing between down-(R-)higgsino with thebino-singlino and wino-triplino states follows. Hence, for large tanβ the down-(R-)higgsinohas appreciable couplings only to staus due to the Yukawa coupling.

The MRSSM-unique λ and Λ parameters are also of relevance for the phenomenologyof charginos and neutralinos. They give additional couplings to the SM-like Higgs bosonobserved at the LHC or to a light singlet H1, present in the light singlet scenario. Large Λare especially necessary to generate the quantum corrections needed for the correct massof the SM-like Higgs boson. This leads to an enhancement of the decay of the wino-triplinoto the SM-like Higgs boson and LSP by the Λ parameters.

In the light singlet scenario there is the additional case where the chargino/neutralinodecay to the SM-like Higgs boson is kinematically forbidden. Then, it might be still possiblethat they decay into the light singlet, even if their respective coupling is suppressed by thedoublet-singlet mixing. For higher neutralino masses the branching ratio to the light singletis sub-dominant to the one to the SM-like Higgs boson.

Summarizing the previous comments, an overview of possible observed signatures ofsets of charginos and neutralinos (listed in the first column) is listed in table 7.3. It givesthe corresponding possible interpretations of signatures in the MSSM (second column)and in the MRSSM (third column). The interpretations are very different, which allows todifferentiate between both models when several fermionic non-SM states are discoveredat the LHC.

A CASE STUDY: COMPARING THE MSSM AND MRSSM USING LHC DATA

In the last section some general understanding of the differences in the LHC signalsbetween the MSSM and MRSSM has been summarized. In the following both modelsare compared for a specific case. The first scenario in tab. 7.3, when two new fermionicparticles are discovered at the LHC, shows that this can be accommodated by the MRSSMand the MSSM. The branching ratios of the states help to disentangle the question ofthe underlying model. The winos of the MSSM couple via gauge couplings and decaydemocratically into all lepton flavors, while for the MRSSM higgsino fields the interactionsare driven by Yukawa couplings leading to final states with third generation SM particles.

For comparison, a scan in the relevant parameter space of the MSSM and MRSSM wasperformed. Motivated by dark matter relic density of section 6.2 the staus are assumed tobe relatively light and the stau mass is varied in both models. The LSP, a bino-like state inthe MSSM and a bino-singlino state in the MRSSM, is assumed to be light with a massof 50 GeV. The lightest chargino and second-lightest neutralino is, as given in tab. 7.3, a

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7.3. Electroweak LHC searches in the MRSSM

experimental signature MSSM MRSSM

1 charged andwino-like states

either up- or down-(R-)higgsino-likestates1 neutral of similar mass

2 charged and- wino-triplino-like states

1 neutral of similar mass

1 charged andhiggsino-like states -

2 neutral of similar mass

2 charged and-

either up- or down-(R-)higgsino-likestates2 neutral of similar mass

2 charged andall states -

3 neutral of similar mass

3 charged and-

wino-triplino-like and either up- ordown-(R-)higgsino-like states2 neutral of similar mass

4 charged and- all states

3 neutral of similar mass

Table 7.4. Possible discovery scenarios at colliders of different sets of particles from theneutralino-chargino sector and the corresponding dominant gauge eigenstatesfor the MSSM and MRSSM. This is assuming the light singlet scenario in theMRSSM, which leads to a light bino-singlino as LSP candidate.

100 200 300 400 500

m 02 , ±

1[GeV]

100

200

300

400

500

mR

[GeV

]

0.10

0.1

0

0.0

5

0.0

5

MSSM

100 200 300 400 500

m 02 , ±

1[GeV]

100

200

300

400

500

mR

[GeV

]

0.01

0.1

0

0.05

0.0

5

MRSSM

Figure 7.3. Exclusion plots in the chargino-neutralino and stau mass plane for the firstscenario of tab. 7.4 in the MSSM (left) and in the MRSSM (right). The violet(blue) region marks the 90% (95%) C.L. exclusion limits given by CheckMATE,each denoted by 1 − C.L..

wino-like state in the MSSM and a down-(R)-higgsino in the MRSSM. Hence, for the MSSMthe wino mass parameter M2 is varied, while for the MRSSM it is the down-(R)-higgsinomass parameter µd . Dark matter direct detection limits 6.3 disfavor low values for theup-(R)-higgsino mass parameter µu.

Taking into account the experimental analyses as described in section 7.2.2, the results

93

7. Direct SUSY searches at the LHC

of the scans in the parameter space are shown in the panels of fig. 7.3. The exclusionbounds are very different between the MSSM and the MRSSM cases. The producedcharginos in the MSSM are wino-like, hence they couple via the same gauge coupling to allstates. All possible decay channels (Higgs, W/Z, staus) contribute then with comparablesize to the total width.

None of the signal regions considered in the LHC analyses is designed to account fora SUSY field decaying into several distinct final states but each is sensitive to a subsetof them, hence the derived limits are rather weak. This is especially striking if the decayto the Higgs boson is dominant as channel in the region where mχ0

2> mχ0

1+ mH . As

described before, it is hard to account for a Higgs boson as decay product of the SUSYstate. In the analyses considered here, charged leptons in the final states are required inlarger multiplicity than normally provided by a Higgs boson decay, which leads to a lowexclusion power. Of course, dedicated analyses taking Higgs bosons in the final stateinto account exist [276, 277] but are not implemented in CheckMATE. Still, the sensitivityto a Higgs boson as decay product is never as good as to a vector boson at a hadroncollider. Therefore, only two rather small regions are excluded: the region around thechargino/neutralino mass of 150–300 GeV and stau mass below 150 GeV and the regionwith a chargino/neutralino mass below, depending on the stau mass.

For the MRSSM the produced electroweak SUSY states are down-(R)-higgsino-like and,due to the large Yukawa coupling and if kinematically accessible, the decay to stau ispreferred. Hence, signal regions relying on events with large multiplicity of taus are verysensitive and yield a large exclusion power. So, the wedge region between the decaythreshold (mχ0

2> mτR + mχ0

1) to around mχ0

2≈ 450 GeV is excluded. However, to the left of

this region, direct decays to staus are not allowed and three-body decays are suppressed.Additionally, the decay products become so soft that they do not pass the cuts imposed bythe analyses. Because of this, it is not possible to exclude that region with the Run 1 dataand analyses strategies.

7.3.2. SLEPTONS

In the MRSSM, the direct production of sleptons at the LHC is comparable to the caseof MSSM. Sleptons can only be produced in pairs via Drell–Yan processes. The maindifference between the MRSSM and the MSSM, the vanishing of the left-right mixing,has no large impact on the LHC bounds as it is usually to be assumed small in theMSSM. In both models, light sleptons decay directly to the corresponding leptons andthe bino(-singlino)-like neutralino LSP. In this scenario, the difference in Dirac or Majorananature of the LSP is of no consequence and, hence, the MSSM exclusion bounds for lightsleptons can be taken directly over to the MRSSM. When the production of sleptons isconsidered, it is usually selectrons and smuons assumed that are meant as their decayproducts can identified easily in the detector.

The ATLAS Collaboration derived the exclusion limits in the (m ˜L,R, mχ0

1) parameter space

from the analyses of the selectron and smuon pair-production processes, see figs. 8 (a)and (b) of Ref. [246] shown in fig. 7.4.† In fig. 7.5 the plots for the case of the MRSSMare depicted. it can be seen that the corresponding bounds are very similar between theMSSM and MRSSM.

†Similar limits are also given by the CMS collaboration [277].

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7.3. Electroweak LHC searches in the MRSSM

[GeV]±1l

~m100 150 200 250 300 350 400

[GeV

]0 1χ∼

m

0

50

100

150

200

250

300

350

)theorySUSYσ1 ±Observed limit (

)expσ1 ±Expected limit (

excludedR

µ∼LEP2

ATLAS = 8 TeVs, -1 Ldt = 20.3 fb∫

All limits at 95% CL

0

1χ∼

±

l0

1χ∼± l→ R

±

l~

R

±l~

)01χ∼

) < m

(±l~

m(

[GeV]±1l

~m100 150 200 250 300 350 400

[GeV

]0 1χ∼

m

0

50

100

150

200

250

300

350

)theorySUSYσ1 ±Observed limit (

)expσ1 ±Expected limit (

excludedR

µ∼LEP2

ATLAS = 8 TeVs, -1 Ldt = 20.3 fb∫

All limits at 95% CL

0

1χ∼

±

l0

1χ∼± l→ L

±

l~

L

±l~

)01χ∼

) < m

(±l~

m(

Figure 7.4. Exclusion plots in the LSP and slepton mass plane for right-handed (left) andleft-handed (right) selectron and smuon. Taken from ATLAS ref. [246], fig. 8(a)and 8(b).

In the normal ordered scenario, left-handed slepton masses below ≈ 300 GeV areexcluded. For the right-handed slepton masses the bound is somewhat weaker (around250 GeV) due to a reduced production rate. Further, the region of small mass differencebetween the sleptons and the LSP is not excluded as the final state particles become softand the analysis cuts are ineffective.

For the light singlet scenario in the MRSSM, where equation (4.11) implies an upper limitof ≈ 55 GeV on the LSP mass, the information from the exclusion limits are as follows:Left-handed slepton masses below ≈ 300 GeV are excluded. For the right-handed sleptonmasses the bound is somewhat weaker (around 250 GeV) due to smaller production crosssection. Also, a small corner at a lower right-handed slepton mass ∼ 100 GeV with the LSPmasses in the range 40-55 GeV is still allowed. With the experimental reach of the Run 1data the direct production of staus with the cross sections predicted by supersymmetrycan not be excluded, see Ref. [278].

7.3.3. CHARGINOS AND NEUTRALINOS

The comparison of the MRSSM with the MSSM and their impact on LHC physics in theprevious sections showed that for the case of direct slepton production no difference can befound. A similar scenario of the MSSM and the MRSSM including charginos and neutralinoswas also investigated and noticeable phenomenological differences were revealed, whichallow a distinction between both models if SUSY is discovered at the LHC.

In the following, some other specific scenarios in the MRSSM are investigated, whichhave no counterpart in the MSSM. As noted before, the benchmark points of the lightsinglet scenario are the basis for this study. The bino-singlino is very light, mχ0

1∼ 50 GeV,

in which case the sensitivity of the LHC analyses is strongest. Besides this, the relevantmass parameters are the Dirac wino-triplino mass MD

W and the two higgsino masses µd,u.Additionally, the exclusion limits depend on the slepton masses influencing strongly thedecay patterns.

Figure 7.6 shows parameter scans and exclusion limits for the case of large slepton

95

7. Direct SUSY searches at the LHC

100 200 300

mL

[GeV]

20

60

100

140

180m

0 1[G

eV]

0.01

0.1

00.0

5

right-handed sleptons

200 300

mL

[GeV]

20

60

100

140

180

m0 1

[GeV

]

0.01

0.1

0

0.05

left-handed sleptons

Figure 7.5. Exclusion plots in the LSP and slepton mass plane for right-handed (left) andleft-handed (right) selectron and smuon. The violet (blue) region marks the90% (95%) C.L. exclusion limits given by CheckMATE, each denoted by 1 − C.L..All other unvaried parameters as for BMP2.

100 200 300 400 500

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Figure 7.6. Exclusion limits in the MRSSM for the chargino and neutralino productionassuming heavy sleptons as a function of the higgsino and wino-triplino masses(left) and of the two higgsino masses (right). The violet (blue) region marksthe 90% (95%) C.L. exclusion limits given by CheckMATE, each denoted by1 − C.L.. All parameters which are not varied in the plots are fixed to the valuesof BMP5.

masses of around 500 GeV, on the left in the µd=µu − MDW plane, and on the right in

the µd − µu plane (with MDW = 500 GeV). In both cases, parameter regions with both

higgsinos heavier than around 350 GeV can not be excluded by Run 1 experimental data.The benchmark point BMP5 is a representative of this generic, viable parameter region inscenario of heavy sleptons.

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Figure 7.7. Exclusion limits in the MRSSM for the chargino and neutralino productionassuming light staus as a function of the two higgsino mass parameters µd,u.The violet (blue) region marks the 90% (95%) C.L. exclusion limits given byCheckMATE, each denoted by 1−C.L.. The left plot has light right-handed sleptonmasses of 100 GeV. In the right plot only the right-handed stau mass is fixed to100 GeV. All other parameters are set to the values of BMP5. Most importantly,the Dirac wino mass is set to MD

W = 500 GeV.

From the plot on the left of fig. 7.6 it is clear that the limit on the higgsino masses isalmost independent of the wino-triplino mass MD

W , excluding masses below around 300 to350 GeV. The limit for MD

W in the case of heavier higgsinos is rather weak, excluding onlymasses upto 150 to 200 GeV. As described earlier, the wino-triplino decays predominantlyinto Higgs bosons and LSP, with the Higgs boson predominantly decaying further into bquarks. The experimental analyses used here are not sensitive to this as there is no signalregion yielding a good exclusion power for such a final state.

Assuming that the wino-triplino mass MDW has no strong influence on the exclusion

bounds of the other parameters, the right plot of fig. 7.6 shows the limits when thehiggsino masses µd and µu are varied independently.

Two additional regions in parameter space at higgsino masses around 150 GeV are notexcluded by the considered LHC searches. The second-lightest neutralino and lightestchargino, which are produced predominantly, have masses just above the sum of LSPand Z/W boson mass. Hence, their decay products become rather soft. This makes isdifficult to separate the signal events from the background of standard on-shell W/Z/Higgsproduction and decay and the analyses cuts are not as efficient as for higher masses. Incontrast, for the very low higgsino masses there is no allowed two-body decay and all decaymodes of the charginos and neutralinos are three-body decays. The kinematic variables ofa three-body decay do not have the signature of a resonance and can be separated froman SM boson decay. Hence, the background for the lepton searches is reduced and theexclusion power enhanced.

The exclusion contours depend on the two (R)-higgsino mass parameters µd–µu inthe scenario of light right-handed staus are presented in fig. 7.7. In the left panel, allleft-handed sleptons are chosen to be heavy, above a mass of 500 GeV, and the masses

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of the right-handed selectron and smuon are set to the right-handed stau mass. In theright plot the scenario, where all the additional sleptons are rather heavy, above 500 GeV, isshown. For both plots there is an interesting non-excluded region with very small µd , belowaround 150 GeV. This is consistent with the discussion for fig. 7.3 and the correspondingnot excluded low-mass region found there. Moreover, when the masses of the charginosand neutralinos become to heavy the signal regions lose their exclusion power, dependingon the on the exact decay patterns at different limits, as described below. From theseconsiderations is becomes clear that in the scenario of light staus two distinct viableparameter regions are viable, one with very small µd and one with larger µd . These tworegions are represented by the two benchmark points BMP4 and BMP6.

The region for µd between around 150 and 400 GeV corresponds to the excludedparameter space in fig. 7.3, in which the decay of the higgsino into stau is dominant, andis excluded. Small differences in the ranges of the exclusion bounds between left plot infigure 7.3 and 7.7 can be attributed to the differences of the mass parameter µd and theactual pole masses mχ0

2.

Additionally, the region with the up-(R)-higgsino mass parameter µu less than roughly300 GeV is excluded, if all right-handed sleptons are light. The mixing of the up-(R)-higgsinowith the singlino-bino allows for it to decay notably into all slepton flavors in the same ratein addition to the decays into Z and Higgs boson. Therefore, the branching fraction intoselectrons and smuons is enhanced which yields a higher efficiency of the experimentalanalyses. This is contrary to the behavior of the down-(R)-higgsino, as described before.Here, large tanβ suppresses mixing to the singlino-bino and the large tau Yukawa couplinggives rise to a predominant decay rate into staus.

In the case of heavy sleptons of the first two generations, the exclusion power on µu

is very weak as the decay to Z and Higgs boson have comparable branching ratios. Asdiscussed before, the signal regions of the considered analyses have a low exclusion powerin the case of large branching factions into Higgs bosons.

7.4. SUMMARY

In this chapter an overview of the MRSSM physics at the LHC with focus on the electroweaksector, namely the sleptons and charginos and neutralinos, has been given. The findingsare summarized in the following.

Should sleptons be produced at the LHC a similar result as in the MSSM can be derivedas shown in fig. 7.5. In general, left-handed sleptons are constrained more strongly thanright-handed ones. Using the results of Run 1 this means an upper limit of 300 GeV can beput on the mass of the left-handed and 250 GeV on the mass of the right-handed sleptons.A region close to the mass degeneracy with the LSP can not be excluded because of thekinematics of the decay.

In the case of charginos and neutralinos the issue is more complex as in general theMRSSM sector contains more degrees of freedom as in the MSSM. Furthermore differ-ences in the coupling structure and the Dirac nature of neutralinos lead to a more complexcomparison. As a result, a clear difference in the LHC phenomenology of MSSM andMRSSM can be found as exemplified by tab. 7.4.

When considering the light singlet scenario and its benchmark points the findings canbe summarized as follows. Several distinct parameter regions still allowed by LHC Run 1

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electroweak searches have been found, illustrated by the three benchmark points BMP4,BMP5 and BMP6. As the light singlet scenario is used as the foundation of this studythese regions all have a light bino-singlino state as LSP in common. The wino-triplino massparameter MD

W plays no important role for the LHC exclusion limits and is chosen to be inthe range of 400–600 GeV for the benchmark points. From dark matter considerations theneed for a light stau arises. Therefore, two regions include a light right-handed stau with amass of around a 100 GeV and a down-higgsino with a low mass of also around 100 GeVor a high mass above 400 GeV. The third region is defined by generally heavy sleptons andhiggsinos with masses above around 300 GeV.

An overview of the LHC physics of the other MRSSM sector is given in sec 7.2.1. Itshows that the MRSSM offers a rich phenomenology to be explored at the LHC if it isindeed a valid description of nature.

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8. EXPLORING THE PARAMETERSPACE

The phenomenology of the MRSSM has been studied from various vantage points inthe previous chapters. The effects of individual parameters on Higgs (chapter 4) andelectroweak precision (chapter 5) observables as well as constraints from dark matter(chapter 6) and LHC (chapter 7) searches have been determined and investigated using thebenchmark scenarios defined in chapter 3.

It is of interest to combine the different experimental constraints and study the interplayof the observables on different parameter combinations. This allows to identify parameterregions which are viable when taking into account all constraints and can be used asfoundation for further studies and to make definite model predictions for new physics.

In the first part of this chapter, the interdependence of the SM-like Higgs and W bosonmass is examined in the normal ordered scenario, this is based on refs. [144, 145]. Next,the parameter space of the light singlet scenario is explored not only taking into accountthe Higgs and W boson masses but also results from dark matter studies. This was firstpresented in ref. [146].

The benchmark points of the normal ordered scenario were introduced in ref. [145]and updated in [144]. There, Higgs and electroweak observables were studied and thebenchmark points represent regions of parameter space favored by these constraints. Theinclusion of dark matter constraints actually disfavors the selected benchmark points asthe bino-singlino is not the dark matter candidate. This will be rectified in the last sectionshowing how the benchmark points BMP1, BMP2 and BMP3 can be adapted to the darkmatter constraints found in chapter 6.

8.1. CONSTRAINING THE NORMAL ORDERED SCENARIO

To obtain the correct mass of the SM-like Higgs boson it has been shown in chapter 4 thatat least some of the superpotential parameters λd , Λd , λu and Λu should in magnitudebe close to unity and a certain mass splitting between the scalar and fermionic sector isrequired.

Effects from the additional MRSSM states on the electroweak precision observablesand the W boson mass are mainly driven by these new superpotential couplings and the

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oblique parameters are sensitive to how close the masses of the superpartners are to themass scale of the electroweak symmetry breaking.

The combined dependency of the SM-like Higgs and W boson masses on several param-eter combinations is shown in figs. 8.1 to 8.3. The observables are given as golden-greencontour regions for the Higgs boson mass and black contour lines for the W boson mass.Investigating different parameter combinations for all three benchmark points allows tosample different tanβ regions. White regions denote regions where tachyonic states inthe mass spectrum appear.

In fig. 8.1 the impact of the Yukawa-like λ and Λ parameter on the Higgs and W bosonmass and their interdependence is shown. It was pointed out in the correspondingchapters how these parameters affect the masses. For mH1 the dependency is clearfrom the tree-level result (4.3) and the contributions from the effective potential (4.7).The mW dependency can be grasp from the S, T and U oblique parameters using theapproximation (5.24) given by eqs. (5.28) to (5.35).

On the left side of fig. 8.1, λu and Λu are varied against each other. The W boson massdependency is given in form of ellipses around a minimal value for these parameters. Thisis consistent with the result of chapter 5. The main contributions to the W boson massare given by the neutralinos and charginos through the T parameter, see eqs. (5.24) andfig. 5.3. The oblique parameter T depends similarly on λu and Λu to the fourth power, seeeqs. (5.31) and (5.32). The minimum in fig. 8.1 is not exact at λu,Λu = 0, but it can be seenfrom fig 5.3 that non-oblique corrections shift it to non-zero values.

The Higgs boson mass depends mainly on Λu, which couples the triplet to the Higgsboson. This can be explained with the quartic dependency shown by the effective potentialcontribution (4.7) and the rather large triplet mass required to ensure a small vT . Thesinglet parameter λu becomes only important for the Higgs boson mass with values aboveunity. Due to the high sensitivity of both masses on the superpotential parameters, theregion where both experimental constraints are satisfied is rather small, but agreementwith experiment is possible.

The right side of fig. 8.1 shows how tanβ influences the relative contributions from up-(λu) and down-type (λd ) parameters. For BMP1 (top plot) with tanβ = 3, λd affects thesize of mH1 depending on its sign and magnitude. With rising tanβ, this effect vanishesand the Higgs boson mass becomes approximately constant with regard to λd for BMP3with tanβ = 40. This originates from the fact that the up-type parameters generally areaccompanied by vu, while the down-type parameters appear together with vd which is neg-ligible for large tanβ. For tanβ 1, λu and Λu give therefore the dominating contributionscompared to λd and Λd . The W boson mass has a similarly stronger dependency on λu, asthe corresponding vev has to be inserted in eq. (5.32).

Fig. 8.2 forms the basis to investigate the dependence of the Higgs and W boson masseson the dimensionful superpotential parameter µu and the Dirac mass terms MD

B and MDW .

On the left side of fig. 8.2, µu and MDW are varied against each other. The Higgs boson

mass has an elliptic dependency on the parameter combination with the maximum forboth parameters at zero. This is understandable from the loop corrections (4.7) dependingon the ratio of Dirac masses to soft breaking masses, which is maximized if µu and MD

Ware close to zero. The vector-like limit for the neutralino and chargino contributions (5.33)can be used to understand the dependency of the W boson mass. For increasing massparameters the actual suppression of the T parameter with large scales is most important.

The right panels of fig. 8.2 highlight the dependency on the Dirac masses MDB and

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MDW . For BMP3 (bottom panel), MD

B is not important for either observable as the singletYukawa-like parameters are close to zero. Any contribution containing MD

B is thereforesuppressed by small prefactors. The dependencies for BMP1 and BMP2 are comparableto each other. The T parameter contributions are reduced for larger Dirac mass terms,see eqs. (5.31) and (5.32), leading to a reduction of mW . The same is true for mH1 , as thelogarithms of the loop contributions (4.7) are reduced for larger fermonic mass parameters.The exact behavior depends on the size of the λ and Λ couplings. For BMP1 both categoriesare of similar size; while λd and λu are larger for BMP2, they are close to zero for BMP3.The shift of the Higgs boson mass maximum away from zero mass parameters for BMP1and BMP2 originates from the cancellation of λuµu with g1MD

B in the tree-level massformula (4.3).

The Higgs boson mass prefers small values for the Dirac masses, except for BMP2, whilethe W boson prediction coincides with experiment for large values of the mass parameters.Still, regions in parameter spaces where both constraints are satisfied can be found. TheYukawa-like superpotential parameters and Dirac masses can be varied against each otherto bring the constraints from the Higgs and the W boson masses in compliance. This isshown by the plots of fig. 8.3. On the left side of the figure, λu and µu are altered andit becomes apparent that it requires negative values for λu to bring both constraints intoagreement. BMP3 is the exception to this as for this benchmark point Λu is already largeenough to achieve the correct Higgs boson mass by its contributions alone. Then, the sizeof the other superpotential couplings like λu needs to be smaller than one so that neitherthe Higgs nor the W boson mass becomes too large. As stated before, mH1 prefers smallerµu while mW has a minimum at rather large values (The position of the minimum dependson the exact value of MD

B , see fig. 8.2).The variation of Λu against MD

W is depicted on right side of fig. 8.3. The Higgs bosonmass shows only a mild dependence on MD

W compared to µu and MDB . This is because

the relevant logarithm in the effective potential contribution (4.7) includes MDW and also

m2T , which is chosen to be large for all benchmark points. Then, a change in MD

W on thescales investigated here has only a small influence. The dependence of mW on MD

W ismore complex. In the regions where the magnitude of Λu is moderate the S parameter canbe as large as the T parameter, see fig. 5.2, and contributes with a minus sign to the Wboson mass (5.24) leading to a reduction of it.

Both mass observables rise steeply with Λu and the region where both constraintsare satisfied is rather narrow. Hence, both masses can be brought into accordance withexperiment by shifting Λu to achieve the correct Higgs boson mass and then adapt MD

W forthe correct mW .

All these figures show that an appropriate range for the input parameters exists to arriveat the experimental value for the W and Higgs boson masses separately, but there are onlycertain regions were both, mH1 and mW , are in simultaneous agreement with experiment.

In these regions, at least one λ or Λ parameter has a magnitude close to −1, whichenhances the loop corrections to obtain the Higgs boson mass. Large scalar masses arerequired to reduce the contributions to the T parameter and, together with moderate Diracmasses, contribute crucially to the effective potential.

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8.2. Constraining the light singlet scenario

8.2. CONSTRAINING THE LIGHT SINGLET SCENARIO

The light singlet scenario exemplified by BMP4, BMP5 and BMP6 provides an interestingphenomenology to be studied at the LHC. The relevant parameters are already constrainedto accommodate existing experimental results. The most important constraints are, as inthe normal ordered scenario, the Higgs and W boson masses. Additionally, dark matter relicdensity and direct detection limits put severe restrictions on certain parameters. Thesefour constraints are summarized in Fig. 8.4. Like in the section before, the SM-like Higgsboson mass is given by the green-yellow color bands, while black full contour lines give theW boson mass. The blue dashed lines show the DM relic density, while the violet regionis excluded with 95% C.L. by direct detection searches. All panels are based on BMP5,except for the bottom-right one, where BMP4 is used.

The model contains a light singlet-like Higgs boson if the parameters mS, MDB and λu

are fixed to comparably small values. The SM-like Higgs boson mass is then increased viamixing at tree level compared to the normal ordered scenario, but positive loop correctionsgoverned by Λu are still required to drive the mass to the experimentally observed value.

Consequently, Λu is fixed by the Higgs mass measurement and MDB and µu are con-

strained due to the mixing with the singlet. As discussed for the normal ordered scenario,the W-boson mass can be explained simultaneously with the SM-like Higgs boson mass.Fig. 8.4 confirms that this is still the case in the light singlet scenario.

The Dirac bino-singlino as the LSP leads to the correct dark matter relic density if otherstates, especially the right-handed stau, are in a suitable mass range. If the LSP mass isclose to half of the Z boson mass all sleptons must be as heavy as for BMP5, else theannihilation would be too effective. This resonance behavior is evident in the top left panelof fig. 8.4. If the LSP mass is not close to half of the Z boson mass, the right-handed staumust be light, as it is for BMP4 and BMP6. As µu and µd appear in the neutralino massmatrix , they affect the LSP mass and, hence, also the relic density via mixing. This isevident in the right column of fig. 8.4. The effect is especially concise as for the underlyingBMP5; a small change in the LSP mass leads to a large change in the relic density.

Null searches for dark matter and negative LHC searches for SUSY states constrain theparameter space further. As shown by the top right of fig. 8.4, m2

qR ;1,2 and µu are correlatedby dark matter searches due to the interference between the contributing processesdescribed in chapter 6. Together with the observed Higgs boson mass, these searchesconstrain both µu and m2

qR ;1,2 significantly. A lower bound on µu can be inferred if the lowerlimit on the squarks masses from LHC searches is included. This bound is not impaired byother parameters of the weak sector like Λu, see the bottom left of fig. 8.4.

The recast of LHC analyses in chapter 7 brought out three interesting parameter regions.They are exemplified by the three benchmark points BMP4, BMP5 and BMP6. Evidently,the region characterized by heavy MD

W , µd and sleptons (BMP5) is still viable. Moreover, asecond region with a small right-handed stau mass of around 100 GeV (BMP6) and a thirdwith light right-handed stau and small µd (BMP4) are also still conceivable. As explainedin chapter 7, these two regions are allowed because the Run 1 LHC searches are notrestrictive for these mass hierarchies. The last region is also not disfavored by the otherconstraints as portrayed by the bottom right plot of fig. 8.4. Parameter regions with rathersmall µu are also allowed by LHC searches, however they are excluded by dark matterconstraints.

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8. Exploring the parameter space

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8.3. The normal ordered scenario and dark matter

8.3. THE NORMAL ORDERED SCENARIO AND DARK MATTER

The benchmark points BMP1, BMP2 and BMP3 have originally been defined withoutconsidering constraints from dark matter. To achieve agreement with dark matter results,further dependencies have to be studied. In the following, these benchmark points areadapted to satisfy all data. Many of the model parameters are then strongly constrained inorder to accommodate the dark matter observables. The modifications can be characterizedas follows:

• The µu and µd parameters of all points are lifted to 650 GeV to satisfy dark matterdirect detection limits.

• The λ superpotential parameters are also affected by DM direct detection and needto be close to zero as is be discussed below.

• To achieve the correct DM relic density, the mass of the bino-singlino is reduced forBMP1 to 170 GeV as annihilation via sleptons is the dominant channel and requiresan LSP which is lighter than 350 GeV.

• The right-handed stau soft mass for BMP1 and slepton soft masses for BMP3 arereduced close to the LSP mass to 200 GeV and 300 GeV, respectively, for an effectiveannihilation process.

• BMP2 is chosen to represent the region of wino-triplino coannihilation with the LSP.• The Λ superpotential parameters are adapted to compensate changes in the Higgs

and W boson mass bringing them near the experimental value.

The adapted BMP1, BMP2 and BMP3 are denoted as BMP7, BMP8 and BMP9, respectively.Tab. 8.1 summaries the exact numerical changes for the three benchmark points. With theplots in fig. 8.5, several aspects of the dark matter constraints are investigated in moredetail. The panels show, as before, the Higgs and W boson mass and, additionally, thedark matter observables, the relic density as blue line and the direct detection exclusion asviolet region, where a red line marks the 95% C.L. exclusion limit. In the top left plot, λu isvaried against µu = µd ; the other parameters are chosen as for BMP7. It is clear that λu

actually has to be small to achieve agreement with the direct detection limit in agreementwith the Higgs boson mass. This is required to constrain the singlino-higgsino mixing, seethe neutralino mass matrix C.26. As known from the light singlet scenario, µu should belarger than 550 GeV. If it becomes too large, tension with Higgs and W boson massesarises and this limits the higgsino mass parameters to the range of roughly 550 to 700 GeVin this example.

For BMP8 co-annihilation of the LSP with the wino-triplinos is used as mechanism torealize the correct dark matter relic density. This is shown in the top right panel of fig. 8.5,where Λu is varied against the relevant mass parameter MD

W . The direct detection exclusionbound marks the region where a charged wino-triplino becomes actually the LSP. This iscompletely disfavored by experiment. The relic density is only close to the experimentalvalue in the region, where the wino-triplino mass is close to the LSP mass as expected fromco-annihilation. The dependency of the relic density on Λu originates from mixing betweenthe wino-triplino and the higgsinos, which it induces in the neutralino mass matrix (C.26).This mixing affects the mass of the state needed for co-annihilation significantly. The Wboson mass is only slightly and the Higgs boson mass is not at all affected by MD

W . It istherefore possible to bring these masses close to their observed value by adjusting Λu afterMD

W is fixed.

109

8. Exploring the parameter space

BMP7 BMP8 BMP9

based on BMP1 BMP2 BMP3

λd , λu 0.0, −0.1 0.3, −0.3 0.15, −0.15Λd , Λu −1.35, −1.35 −1.0, −1.25 −1.0, −1.15MD

B 170 450 250MD

W 500 446 500µd , µu 650, 650m2

E;1,2, m2E;3 10002, 2002 10002, 10002 3002, 3002

vS 0.86 0.25 −0.43vT −1.26 −1.53 −1.19m2

Hd3182 5992 10382

m2Hu

−7762 −7632 −7502

Table 8.1. Shown are the changes to the benchmark points of the normal ordered scenariofrom tab. 3.1 to arrive at the benchmark points in agreement with all consideredobservables. Dimensionful parameters are given in GeV or GeV2, as appropriate.The last part shows the derived parameters.

For BMP9, exemplified in the bottom panel, annihilation through the right-handed sleptonchannel is used to achieve the observed DM relic density. The unified slepton massesare changed to be close to the LSP mass but in a range not excluded by LHC data. Thisfixes the dark matter constraints independently of Λu for this annihilation scenario. TheDM direct detection exclusion limit is only of relevance if the stau mass is so low that itbecomes the LSP, which is clearly disfavored. Then, Λu can be fixed to a value so that thecorrect Higgs and W boson masses are achieved.

110

8.3. The normal ordered scenario and dark matter

1.5 1.0 0.5 0.0 0.5 1.0 1.5

u =- d

400

500

600

700

800

900

1000

1100

1200

u=

d[G

eV]

80.4

10

80.4

30

80.4

30

0.1

2

0.1

2

0.1

4

0.1

4

2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0

u

400

450

500

550

600

650

700

MD W

[GeV

]

80.3

80

80.3

90

80.3

90

80.4

10

80.4

10

80.4

30

80.4

30

0.05

0.12

0.50

1.0

0

2.0 1.5 1.0 0.5 0.0 0.5 1.0

u

200

400

600

800

1000

ml R

[GeV

]

80.3

80

80.3

80

80.3

90 8

0.3

90

80.4

10

80.4

10

80.4

30 8

0.4

30

80.4

80 8

0.4

80

0.05

0.12 0.120.12

0.50

1.00

50 100 117 122 128 133 150 200

mH2[GeV]

0.00 0.05 1.00

p value for direct detection

Figure 8.5. Shown are the same observables as in fig.8.4. The unvaried parameters areset to the benchmark points BMP7 (top, left), BMP8 (top, right) and BMP9(bottom).

111

8. Exploring the parameter space

8.4. SUMMARY

All the constraints presented in earlier chapters have been brought together in this chapterto study the parameter space of the MRSSM and the interplay of the different observablesin detail. It has been shown that each experimental input imposes non-trivial bounds onthe parameter space. When studying the correlations of different constraints, the analysisestablishes that concrete regions are viable after taking all constraints into consideration.The preferred parameter regions can be characterized as follows:

• The LSP and hence DM candidate is a Dirac bino-singlino. If no coannihilation withother neutralinos and charginos takes place, it is lighter than 300 GeV, requiring atleast the right-handed stau to be close in mass. Including coannihilation effects themass can go up to at least 500 GeV; further study is required to verify an extendedrange. The bino-triplino is required to be close in mass for coannihilation putting abound on MD

W . The limit on the LSP mass provides the bound on the Dirac mass MDB .

• It is possible to have a singlet-like Higgs boson lighter than the SM-like Higgs bosonin the MRSSM. The singlet soft breaking mass m2

S must than be below ≈ 100 GeV.The Dirac mass parameter MD

B also enters the singlet mass prediction and is requiredto be below ≈ 55 GeV. If no light singlet is needed as prediction of the model thesebounds are removed.

• For tanβ 1, the superpotential coupling Λu is required to be close to −1. If tanβis close to one, this is also true for Λd . This size is required to achieve the correctSM-like Higgs boson mass. The W boson mass does not allow larger magnitudes asit would receive too large contributions.

• The triplet soft breaking mass m2T needs to be sizable to reduce the triplet vev

vT and hence the tree-level contribution to mW . Furthermore, a certain hierarchybetween the scalar mass mT and the fermionic mass MD

W is necessary for the loopcontributions to the Higgs boson mass to be large enough.

• The superpotential couplings λd and λu are required to be close to zero to ensure areduced higgsino content of the LSP. In the light singlet scenario this also prohibitstoo large singlet-doublet mixing.

• The superpotential mass parameter µu is restricted by dark matter direct searchesto be larger than ≈ 550 GeV. This also satisfies the Run 1 LHC limits. To arrive atthe correct Higgs boson mass, µu must not be too large, roughly above one TeV. Fortanβ ≈ 1 these restrictions are also true for µd ; it is unconstrained for larger valuesof tanβ.

• The squark and scalar octet masses as well as the Dirac gluino mass are constrainedby direct LHC searches, which goes beyond the extent of this work, but see forexample refs [205, 250]. The squark masses are also restricted indirectly by darkmatter searches and the exact limit depends on the size of µu and µd .

• All other electroweak scalar soft breaking masses need to be sizable enough toescape current collider limits. They may still lead to states accessible by extendedLHC runs.

The status of all benchmark points with regard to the observables studied within this thesisis given in tab. 8.2. Each of BMP4, BMP5, BMP6, BMP7, BMP8 and BMP9 represents aviable parameter region and shows a distinct phenomenology also accessible by the LHC.Therefore, not to put too fine a point on it, the MRSSM provides meaningful predictions

112

8.4. Summary

regarding the mass spectra and interactions of new states testable at current and futureexperiments.

113

8.E

xplo

ring

the

para

met

ersp

ace

BMP1 BMP2 BMP3 BMP4 BMP5 BMP6 BMP7 BMP8 BMP9

mH1 125.2 125.6 125.4 100 94 95 126.1 125.9 123.7mH2 897 937 1245 126.1 125.6 125.8 906 948 1242mW 80.396 80.382 80.386 80.384 80.391 80.40 80.400 80.384 80.391mχ0

1415 413 251 49.8 43.9 29.3 251 452 251

mχ02

420 422 408 132 401 422 535 479 533mτR 1003 1006 998 128 1000 106 202 1003 307

passes HiggsBounds X X X X X X X X XΩh2 0.02 0.02 2.11 0.12 0.09 0.13 0.14 0.14 0.11

HiggsSignals p-value 0.79 0.80 0.82 0.77 0.78 0.73 0.79 0.78 0.81DM p-value < 10−10 < 10−10 < 10−10 0.95 0.46 0.18 0.55 0.95 0.99LHC CLs 0.89 0.99 0.50 0.74 0.41 0.30 0.91 0.99 0.82

Table 8.2. Given are the relevant observables for all benchmark points defined in this thesis. The first part gives the important masses includingthose necessary for dark matter constraints, In the middle section the passing of all of the HiggsBounds limits is denoted and the valuefor the relic density is given. The last three rows give the p-values for the Higgs observables from HiggsSignals and respective directsearches, DM and LHC (actually CLs).

114

9. CONCLUSIONS

The MRSSM is a supersymmetric extension of the SM. In contrast to the minimal super-symmetric extension, the MSSM, it also contains R-symmetry. This additional symmetryleads to distinct differences between the predictions of both models. While the physicsof the MSSM has been extensively studied by many groups, the phenomenology of theMRSSM is not as well understood. Within this thesis the effort to rectify the situation inseveral directions has been described.

One of the most important predictions of every SUSY model is the mass of the SM-likeHiggs boson. It has been shown that the MRSSM can accommodate the experimentalvalue for the mass of the recently discovered candidate for the Higgs boson. This is viableeven as relevant mechanisms of the MSSM are not available. Important novel contributionsarise from the extended field content and coupling structure of the model. The additionalHiggs bosons and their superpartners, coupled by the new Yukawa-like parameters λd , λu,Λd and Λu, provide quantum corrections to raise the mass prediction of the Higgs bosonmass to the measured value.

With the discovery of the candidate for an SM-like Higgs boson at the LHC additionalmeasurements regarding its properties, like production cross section and decay patterns,become available. In conjunction with the wealth of information from previous experimentsthis allows to constrain the extended Higgs sector of the MRSSM and has been done inthis work. This analysis is especially interesting for scenarios with a phenomenology notreplicable by the MSSM. In the MRSSM, for example, this is the case if the singlet Higgsboson is lighter than the discovered resonance. This case has been studied extensivelyhere and proven to be a viable scenario with a very distinct phenomenology.

A plethora of experimental data is available from studies of the electroweak sector.Extending the field content of the SM, as every SUSY model does, generally influenceselectroweak precision observables. The mass of the W boson is a prediction of thecalculation conducted here and relevant contributions can be parametrized by the precisionobservables. Within this work these new effects were analyzed in detail and regions ofparameter space still allowed by the constraints identified.

Astrophysical evidence of dark matter is one of the major hints for physics beyond theStandard Model and it is expected that a SUSY model provides a dark matter particle.R-symmetry leads to a lightest stable supersymmetric state in the MRSSM which is a validcandidate. It has been shown within this work that the bino-singlino as lightest neutralinocan satisfy constraints from the dark matter relic density over a wide mass range, from

115

9. Conclusions

as low as 20 GeV up to more than 500 GeV. Relevant effects to achieve this require asecond-lightest SUSY state close in mass. This can be either a stau needed for annihilationto taus or a wino-triplino if coannihilation effects are important. Furthermore, bounds fromdirect detection experiments need to be included. This sets indirect bounds on the massesof higgsinos above 600 GeV and squarks of the first generation above 1.5 TeV.

Considering all the previously discussed experimental evidence the MRSSM is notexcluded and still viable. The non-trivial bounds on the parameter space of the MRSSMallow for meaningful predictions and tests of the model by current experiments.

The most direct tests of new physics available can be done at the LHC experiments byproducing new particles in the high-mass regime. The MRSSM contains several states thatcould appear as such resonances. In the latter part of this thesis the electroweak sector ofthe MRSSM, the sleptons, neutralinos and charginos, has been studied in detail. For thisstudy, results from Run 1 of the LHC have been used and limits on signal rates interpretedin the context of the MRSSM. Then, a comparison to the interpretations within the MSSM,usually done by the experimental collaborations, is possible. Some similarities, especiallyfor the sleptons, can be found, but distinct differences are evident for the charginos andneutralinos. These changes stem from a variation of the couplings and the Dirac natureof the MRSSM neutralinos. This leads to different exclusion limits on the masses of newstates.

The comprehensive study of the MRSSM phenomenology within this thesis has shownthat the MRSSM continues to be a valid description of nature. It has been possible toconstrain the parameter space of the model in a meaningful way and make predictionsfor current experiments. Even as many different experimental inputs have been used toachieve this results, several further avenues of research are still unexplored.

The study of dark matter can be expanded. It is not completely clear, in what mass rangeof the dark matter state coannihilation fails as the relevant mode to obtain the correct relicdensity. Also, the case of two-component dark matter containing the lightest neutralinoand R-Higgs boson may be a valid scenario in the MRSSM leading to an interestingphenomenology distinct from the MSSM. It has to be checked if the dark matter relicdensity can be predicted correctly and how direct detection searches are affected in thisscenario.

One of the main predictions of the MRSSM compared to the MSSM is the Dirac natureof the neutralinos and gluino. The appearance of several heavy Dirac fermions at the LHCis a strong indication for the realization of the MRSSM in nature. With additional fermionicdegrees of freedom SUSY predicts also new bosonic states. Hence, the extended Higgssector of the MRSSM also provides candidates for scalar resonances at the LHC. The SUSYQCD sector is affected by R-symmetry due to the novel Dirac nature of the gluino and theprediction of sgluons, scalar color octets. If accessible, squarks and gluinos of the MRSSMmay be produced in large abundance at the LHC. It is known that quantum effects to therelevant processes need to be included for a meaningful prediction. This has not been donebefore for the MRSSM and is of high interest as the Dirac gluino may lead to significantdifferences when comparing results from the MSSM and MRSSM.

116

A. TOOLS AND INPUT PARAMETERS

A.1. CONVENTIONS

The natural unit system is used in this work setting c = ~ = 1. The metric tensor is chosenin the time-like convention

gµν = diag(1, −1, −1, −1) . (A.1)

The anti-symmetric tensors on the spinor spaces are

εαβ =

(0 −11 0

), εαβ =

(0 1

−1 0

)(A.2)

with εαβεβγ = δαγ .

A.2. TOOL OVERVIEW

For a comprehensive summary, an overview of all computer codes used to arrive at theresults presented within this thesis is given in the following.

• The Mathematica [279] package SARAH-4.8.5 [114–118] is used to generate all re-quired tree-level expressions for the MRSSM. Furthermore, the source code forSPheno-3.3.8 [106, 119] including Beta functions up to two-loop order as well as allone-loop self-energies for the MRSSM is available through SARAH. It also contains theimplementation of a two-loop mass calculation for the Higgs bosons, see refs. [280–282]. The output of the mass spectrum generator is given as SLHA-2 spectrumfile [120, 121].

• The generated SPheno code has been adapted to be in agreement with the equationsof this thesis. This affects the files SugraRuns_MRSSM.f90 and LoopCouplings_MRSSM.f90where the variable ρ has to be replaced by 1/ (1 − ∆ρ) in the calculation of the Wboson mass to match eqs. (5.5) and (5.10) where appropriate. Moreover, effectsfrom the light singlet Higgs boson are taken into account by implementation of thereplacement given by eq. (5.14) in the file SugraRuns_MRSSM.f90.

• The results for self-energies, especially the Higgs boson self-energies, have beencrosschecked using the Mathematica package FeynArts-3.7 [124] together with the

117

A. Tools and Input parameters

package FormCalc-8.4 [125, 126]. This calculation relies on a FeynArts model filewith tree-level vertices of the MRSSM generated by SARAH.

• The SARAH/ SPheno code has been checked against results from FlexibleSUSY [122,123]. This is an useful check for correct RGE running and one-loop self-energies.Agreement up to implementational differences was achieved.

• The programs HiggsBounds-4.2.2 and HiggsSignals-1.4.1 [188–193] can be inter-faced to SPheno by setting the appropriate flags in the SPheno input file: In the blockSPhenoInput the flags 11 and 76 are required to be set to 1. The file for the massuncertainties MHall_uncertainties.dat was created by hand. The used options forHiggsBounds and HiggsSignals are given in section 4.3.

• The dark matter study has been done using MicrOMEGAs-4.1.8 [220–224] based onCalcHEP [225, 226] together with LUXcalc-1.0.1 [227]. For this study, the SARAH gen-erated CalcHEP model file was used. Also, the provided CalcOmega_with_DDetection.cppsource file was adapted to include the LUXcalc C++ header and the code given insection A.2.1 was appended to link MicrOMEGAs and LUXcalc. It also takes the Diracnature of the LSP into account.

• Simulation of MRSSM LHC physics was done using Herwig++-2.7 [270, 271, 283]by interfacing the MRSSM UFO [269] model files created by SARAH. The productionof electroweak SUSY states at the LHC has been simulated at leading order andcompared with 8 TeV data using CheckMATE-1.2.0 [264]. The cross sections calculatedby Herwig++ were checked against MadGraph5 _ aMC@NLO [275] and agree up todetails of implementation.

• A framework was written in the Python programming language [284–286] that linksall the tools and allows for scans in the parameter space of the model. It is thefoundation for the research within this thesis.

A.2.1. MICROMEGAS AND LUXCALC INTERFACE

double b , mu, N, lnL ;double s [ 2 ] ;double M, xpSI , xnSI , xpSD , xnSD , GpSI , GnSI ,GpSD,GnSD, fp , fn , ap , an ,

sigmapSI , sigmanSI , sigmapSD , sigmanSD ;LUXCalc_ In i t ( f a l s e ) ;

f o r ( i n t n = 0; n < 2; n++) LUXCalc_SetWIMP_mG (Mcdm, 2*pA0 [ n ] , 2*nA0 [ n ] ,

2*pA5 [ n ] , 2*nA5 [ n ] ) ;

LUXCalc_GetWIMP_mfa (M, fp , fn , ap , an ) ;LUXCalc_GetWIMP_mG (M, GpSI , GnSI ,GpSD,GnSD ) ;LUXCalc_GetWIMP_msigma (M, sigmapSI , sigmanSI ,

sigmapSD , sigmanSD ) ;

/* Do ra te c a l c u l a t i o n s . Af te r any change to the WIMP or ha loparameters , perform the ra te c a l c u l a t i o n s necessary f o rs igna ls , l i k e l i h o o d s , and / or maximum gap s t a t i s t i c s . * /

LUXCalc_CalcRates ( ) ;

118

A.3. Numerical values of SM input parameters

s [ n ] = LUXCalc_Signal ( ) ;

cout << s [ 0 ] << " " << s [ 1 ] << endl ;b = LUXCalc_Background ( ) ;mu = 0.5* s [ 0 ] + 0.5* s [ 1 ] + b ;

N = LUXCalc_Events ( ) ;

lnL = −mu + N* log (mu) − lgamma (N+ 1 . 0 ) ;

std : : cout << lnL << endl ;std : : cout << LUXCalc_LogLikel ihood () < < std : : endl ;std : : cout << LUXCalc_LogPValue () < < std : : endl ;

A.3. NUMERICAL VALUES OF SM INPUT PARAMETERS(αMS(mZ )

)−1= 127.940

(αMS

S (mZ ))−1

= 118.5 Gµ = 1.1663787 × 10−5 GeV−2

mt = 173.34 GeV mMSc (mMS

c ) = 1.275 GeV mMSc (2 GeV) = 2.3 MeV

mMSb (mMS

b ) = 4.18 GeV mMSs (2 GeV) = 2.3 MeV mMS

d (2 GeV) = 2.3 MeVmτ = 1.776 GeV mµ = 105 MeV me = 0.511 MeV

119

B. MRSSM RENORMALIZATIONGROUP FUNCTIONS

In the following, the one-loop (β(1)i ) and two-loop (β(2)

i ) renormalization group beta functionsof the parameters of the MRSSM are given. The equations are taken from the outputof SARAH calculated following refs. [287, 288]. The β functions are given without therespective loop prefactors (16π) and (16π)2.

B.1. BETA FUNCTIONS OF GAUGE COUPLINGS

β(1)g1

=365

g31 (B.1)

β(2)g1

=15

g31

(2085

g21 + 36g2

2 + 88g23 − 6|λd |2 − 6|λu|2 − 9|Λd |2 − 9|Λu|2 − 14 Tr

(|Yd |2

)− 18 Tr

(|Ye|2

)− 26 Tr

(|Yu|2

))(B.2)

β(1)g2

=4g32 (B.3)

β(2)g2

=g32

(125

g21 + 56g2

2 + 24g23 − 2|λd |2 − 2|λu|2 − 7|Λd |2 − 7|Λu|2 − 6 Tr

(|Yd |2

)− 5 Tr

(|Ye|2

)− 6 Tr

(|Yu|2

))(B.4)

β(1)g3

=0 (B.5)

β(2)g3

=g33

(115

g21 + 68g2

3 + 9g22 − 4 Tr

(|Yd |2

)− 4Tr

(|Yu|2

))(B.6)

B.2. BETA FUNCTIONS OF SUPERPOTENTIAL PARAMETER

B.2.1. TRILINEAR PARAMETERS

β(1)Yd

=3Yd |Yd |2 + Yd |Yu|2

+ Yd

(3 Tr

(|Yd |2

)+ Tr

(|Ye|2

)−

715

g21 − 3g2

2 −163

g23 +

32

|Λd |2 + |λd |2)

(B.7)

β(2)Yd

=45

g21Yd |Yu|2 − |λu|2Yd |Yu|2 −

32

|Λu|2Yd |Yu|2 − 4|Yd |4Yd

121

B. MRSSM renormalization group functions

− 2Yd |Yu|2|Yd |2 − 2Yd |Yu|4

+ Yd |Yd |2(

− 3|λd |2 − 3Tr(

|Ye|2)

+ 6g22 − 9Tr

(|Yd |2

)+

45

g21 −

92

|Λd |2)

− 3Yd |Yu|2Tr(

|Yu|2)

+ Yd

(1561450

g41 + g2

1g22 +

332

g42 +

89

g21g2

3 + 8g22g2

3 +128

9g4

3 + 6g22 |Λd |2 − 3|λd |4 −

154

|Λd |4

− |λd |2(

2|λu|2 + 3|Λd |2)

−32

|Λu|2|Λd |2 −25

g21Tr(

|Yd |2)

+ 16g23Tr(

|Yd |2)

+65

g21Tr(

|Ye|2)

− 9Tr(

|Yd |4)

− 3Tr(

|Yd |2|Yu|2)

− 3Tr(

|Ye|4))

(B.8)

β(1)Ye

=3Ye|Ye|2 + Ye

(3Tr(

|Yd |2)

+ Tr(

|Ye|2)

−95

g21 − 3g2

2 +32

|Λd |2 + |λd |2)

(B.9)

β(2)Ye

= − 4Ye|Ye|4 + Ye|Ye|2(

− 3|λd |2 − 3Tr(

|Ye|2)

+ 6g22 − 9Tr

(|Yd |2

)−

92

|Λd |2)

+ Ye

(72950

g41 +

95

g21g2

2 +332

g42 + 6g2

2 |Λd |2 − 3|λd |4 −154

|Λd |4 − |λd |2(

2|λu|2 + 3|Λd |2)

−32

|Λu|2|Λd |2 −25

g21Tr(

|Yd |2)

+ 16g23Tr(

|Yd |2)

+65

g21Tr(

|Ye|2)

− 9Tr(

|Yd |4)

− 3Tr(

|Yd |2|Yu|2)

− 3Tr(

|Ye|4))

(B.10)

β(1)Yu

=3|Yu|2Yu + Yu|Yd |2 + Yu

(3Tr(

|Yu|2)

−1315

g21 − 3g2

2 −163

g23 +

32

|Λu|2 + |λu|2)

(B.11)

β(2)Yu

= +25

Yu|Yu|2(

g21 + 6g2

2 − 3|λu|2 −92

|Λu|2)

− 2Yu|Yd |4 − 2Yu|Yd |2|Yu|2 − 4|Yu|4Yu

+ Yu|Yd |2(

− 3Tr(

|Yd |2)

+25

g21 −

32

|Λd |2 − |λd |2 − Tr(

|Ye|2))

− 9|Yu|2YuTr(

|Yu|2)

+ Yu

(2977450

g41 + g2

1g22 +

332

g42 +

13645

g21g2

3 + 8g22g2

3 +1289

g43 + 6g2

2 |Λu|2 − 2|λu|2|λd |2

− 3|λu|4 − 3|Λu|2|λu|2 −32

|Λu|2|Λd |2 −154

|Λu|4 +45

g21Tr(

|Yu|2)

+ 16g23Tr(

|Yu|2)

− 3Tr(

|Yd |2|Yu|2)

− 9Tr(

|Yu|4))

(B.12)

β(1)Λd

=Λd

(2|λd |2 + 4|Λd |2 + |Λu|2 + 3Tr

(|Yd |2

)+ Tr

(|Ye|2

)−

35

g21 − 7g2

2

)(B.13)

β(2)Λd

= −1

10Λd

(− 45g4

1 − 18g21g2

2 − 485g42 − 6g2

1 |Λu|2 + 10g22 |Λu|2 + 60|λd |4 + 105|Λd |4

+ 20|λu|4 + 30|Λu|4 + 4g21Tr(

|Yd |2)

− 160g23Tr(

|Yd |2)

− 12g21Tr(

|Ye|2)

+ 10|λd |2(

3Tr(

|Yd |2)

+ 4|λu|2 + 8|Λd |2 + Tr(

|Ye|2))

+ |Λd |2(

− 110g22 + 25Tr

(|Ye|2

)+ 30|Λu|2 − 6g2

1 + 75Tr(

|Yd |2))

+ 30|Λu|2Tr(

|Yu|2)

+ 90Tr(

|Yd |4)

+ 30Tr(

|Yd |2|Yu|2)

+ 30Tr(

|Ye|4))

(B.14)

β(1)Λu

=Λu

(2|λu|2 + |Λd |2 + 3Tr

(|Yu|2

)+ 4|Λu|2 − 7g2

2 −35

g21

)(B.15)

β(2)Λu

= −1

10Λu

(− 45g4

1 − 18g21g2

2 − 485g42 − 6g2

1 |Λd |2 + 10g22 |Λd |2 − 6g2

1 |Λu|2 − 110g22 |Λu|2

+ 60|λu|4 + 30|Λd |4 + 20|λd |2(

2|λu|2 + |Λd |2)

+ 30|Λu|2|Λd |2 + 105|Λu|4

122

B.2. Beta functions of superpotential parameter

+ 30|Λd |2Tr(

|Yd |2)

+ 10|Λd |2Tr(

|Ye|2)

− 8g21Tr(

|Yu|2)

− 160g23Tr(

|Yu|2)

+ 75|Λu|2Tr(

|Yu|2)

+ 10|λu|2(

3Tr(

|Yu|2)

+ 8|Λu|2)

+ 30Tr(

|Yd |2|Yu|2)

+ 90Tr(

|Yu|4))

(B.16)

β(1)λd

=λd

(2|λu|2 + 3|Λd |2 + 4|λd |2 + 3Tr

(|Yd |2

)−

35

g21 − 3g2

2 + Tr(

|Ye|2)

(B.17)

β(2)λd

= −1

10λd

(− 45g4

1 − 18g21g2

2 − 165g42 − 120g2

2 |Λd |2 + 100|λd |4 + 40|λu|4 + 75|Λd |4

+ 30|Λu|2|Λd |2 + 4g21Tr(

|Yd |2)

− 160g23Tr(

|Yd |2)

+ 45|Λd |2Tr(

|Yd |2)

− 12g21Tr(

|Ye|2)

+ 15|Λd |2Tr(

|Ye|2)

+ 2|λd |2(

15Tr(

|Ye|2)

+ 20|λu|2 − 30g22 + 45Tr

(|Yd |2

)+ 60|Λd |2 − 6g2

1

)− 12|λu|2

(5g2

2 − 5|Λu|2 − 5Tr(

|Yu|2)

+ g21

)+ 90Tr

(|Yd |4

)+ 30Tr

(|Yd |2|Yu|2

)+ 30Tr

(|Ye|4

))(B.18)

β(1)λu

=λu

(2|λd |2 + 4|λu|2 − 3g2

2 + |Λu|2 + Tr(

|Yu|2)

−35

g21

))(B.19)

β(2)λu

= −1

10λu

(− 45g4

1 − 18g21g2

2 − 165g42 − 120g2

2 |Λu|2 + 40|λd |4 + 100|λ2u| + 30|Λu|2|Λd |2

+ 75|Λu|4 + 4|λd |2(

10|λu|2 − 15g22 + 15|Λd |2 + 15Tr

(|Yd |2

)− 3g2

1 + 5Tr(

|Ye|2))

− 6|λu|2(

10g22 − 15Tr

(|Yu|2

)− 20|Λu|2 + 2g2

1

)− 8g2

1Tr(

|Yu|2)

− 160g23Tr(

|Yu|2)

+ 45|Λu|2Tr(

|Yu|2)

+ 30Tr(

|Yd |2|Yu|2)

+ 90Tr(

|Yu|4))

(B.20)

B.2.2. BILINEAR PARAMETERS

β(1)µD

= 2µD|λd |2 − 3g22µD + 3µD|Λd |2 + 3µDTr

(|Yd |2

)−

35

g21µD + µDTr

(|Ye|2

)(B.21)

β(2)µD

=110µD

(45g4

1 + 18g21g2

2 + 165g42 − 60|λu|4 − 75|Λd |4 − 4g2

1Tr(

|Yd |2)

+ 160g23Tr(

|Yd |2)

+ 15|Λd |2(

− 2|Λu|2 − 3Tr(

|Yd |2)

+ 8g22 − Tr

(|Ye|2

))+ 12g2

1Tr(

|Ye|2)

− 10|λd |2(

3Tr(

|Yd |2)

+ 4|λu|2 + 6|Λd |2 + Tr(

|Ye|2))

− 90Tr(

|Yd |4)

− 30Tr(

|Yd |2|Yu|2)

− 30Tr(

|Ye|4))

(B.22)

β(1)µU

= 2µU |λu|2 −35µU

(5g2

2 − 5|Λu|2 − 5Tr(

|Yu|2)

+ g21

)(B.23)

β(2)µU

=110µU

(45g4

1 + 18g21g2

2 + 165g42 + 120g2

2 |Λu|2 − 40|λd |2|λu|2 − 60|λu|4 − 30|Λu|2|Λd |2

− 75|Λu|4 + 8g21Tr(

|Yu|2)

+ 160g23Tr(

|Yu|2)

− 45|Λu|2Tr(

|Yu|2)

− 30|λu|2(

2|Λu|2 + Tr(

|Yu|2))

− 30Tr(

|Yd |2|Yu|2)

− 90Tr(

|Yu|4))

(B.24)

123

B. MRSSM renormalization group functions

B.3. BETA FUNCTIONS OF SOFT BREAKING PARAMETERS

B.3.1. BILINEAR PARAMETER

β(1)Bµ =

110

Bµ(

− 6g21 − 30g2

2 + 10|λd |2 + 10|λu|2 + 15|Λd |2 + 15|Λu|2 + 30Tr(

|Yd |2)

+ 10Tr(

|Ye|2)

+ 30Tr(

|Yu|2))

(B.25)

β(2)Bµ =

120

Bµ(

90g41 + 36g2

1g22 + 330g4

2 + 120g22 |Λd |2 + 120g2

2 |Λu|2 − 60|λd |4 − 60|λu|4

− 75|Λd |4 − 20|λd |2(

3|λd |2 + 4|λu|2)

− 60|Λu|2|λu|2 − 60|Λd |2|Λu|2 − 75|Λu|4

− 8g21Tr(

|Yd |2)

+ 320g23Tr(

|Yd |2)

+ 24g21Tr(

|Ye|2)

+ 16g21Tr(

|Yu|2)

+ 320g23Tr(

|Yu|2)

− 180Tr(

|Yd |4)

− 120Tr(

|Yu|2|Yd |2)

− 60Tr(

|Ye|4)

− 180Tr(

|Yu|4))

(B.26)

B.3.2. DIRAC MASS PARAMETERS

β(1)MB

D=

25

MBD

(18g2

1 + 5|λd |2 + 5|λu|2)

(B.27)

β(2)MB

D= −

125

MBD

(100|λd |4 + 100|λu|4 − 50|λd |2

(3g2

2 − 3|Λd |2 − 3Tr(

|Yd |2)

− Tr(

|Ye|2))

− 150|λu|2(

− |Λu|2 − Tr(

|Yu|2)

+ g22

)+ g2

1

(130Tr

(|Yu|2

)− 180g2

2 − 208g21 − 440g2

3

+ 45|Λd |2 + 45|Λu|2 + 70Tr(

|Yd |2)

+ 90Tr(

|Ye|2)))

(B.28)

β(1)MW

D= MW

D

(|Λd |2 + |Λu|2

)(B.29)

β(2)MW

D=

15

MWD

(12g2

1g22 + 440g4

2 + 120g22g2

3 + 3g21 |Λd |2 − 40g2

2 |Λd |2 + 3g21 |Λu|2 − 40g2

2 |Λu|2

− 15|Λd |4 − 10|λd |2(

|Λd |2 + g22

)− 15|Λu|4 − 10|λu|2

(|Λu|2 + g2

2

)− 30g2

2Tr(

|Yd |2)

− 15|Λd |2Tr(

|Yd |2)

− 10g22Tr(

|Ye|2)

− 5|Λd |2Tr(

|Ye|2)

− 30g22Tr(

|Yu|2)

− 15|Λu|2Tr(

|Yu|2))

(B.30)

β(1)MO

D= −6g2

3MOD (B.31)

β(2)MO

D=

15

g23MO

D

(11g2

1 − 20Tr(

|Yd |2)

− 20Tr(

|Yu|2)

+ 45g22 + 520g2

3

)(B.32)

B.3.3. SCALAR SOFT BREAKING MASS PARAMETERS

σ1,1 =

√35

g1

(− 2Tr

(m2

uR

)− Tr(

m2˜L

)− m2

hd− m2

Ru+ m2

hu+ m2

Rd

+ Tr(

m2dR

)+ Tr

(m2

eR

)+ Tr

(m2

qL

))(B.33)

σ2,11 =1

10g2

1

(2Tr(

m2dR

)+ 3Tr

(m2

˜L

)+ 3m2

hd+ 3m2

hu+ 3m2

Rd+ 3m2

Ru

+ 6Tr(

m2eR

)+ 8Tr

(m2

uR

)+ Tr

(m2

qL

))(B.34)

124

B.3. Beta functions of soft breaking parameters

σ3,1 =1

201√15

g1

(− 9g2

1m2hd

− 45g22m2

hd+ 9g2

1m2hu

+ 45g22m2

hu+ 9g2

1m2Rd

+ 45g22m2

Rd− 9g2

1m2Ru

− 45g22m2

Ru− 30

(m2

Rd+ m2

hd

)|λd |2 + 30

(m2

Ru− m2

hu

)|λu|2

+ 45m2hd

|Λd |2 − 45m2Rd

|Λd |2 − 45m2hu

|Λu|2 + 45m2Ru

|Λu|2 + 4g21Tr(

m2dR

)+ 80g2

3Tr(

m2dR

)+ 36g2

1Tr(

m2eR

)− 9g2

1Tr(

m2˜L

)− 45g2

2Tr(

m2˜L

)+ g2

1Tr(

m2qL

)+ 45g2

2Tr(

m2qL

)+ 80g2

3Tr(

m2qL

)− 32g2

1Tr(

m2uR

)− 160g2

3Tr(

m2uR

)+ 90m2

hdTr(

|Yd |2)

+ 30m2hd

Tr(

|Ye|2)

− 90m2hu

Tr(

|Yu|2)

− 60Tr(

|Yd |2m2*d

)− 30Tr

(Ydm2*

q Y †d

)− 60Tr

(|Ye|2m2*

e

)+ 30Tr

(Yem2*

l Y †e

)+ 120Tr

(|Yu|2m2*

u

)− 30Tr

(Yum2*

q Y †u

))(B.35)

σ2,2 =12

(3Tr(

m2qL

)+ 4m2

T + m2hd

+ m2hu

+ m2Rd

+ m2Ru

+ Tr(

m2˜L

))(B.36)

σ2,3 =12

(2Tr(

m2qL

)+ 6m2

O + Tr(

m2dR

)+ Tr

(m2

uR

))(B.37)

β(1)m2

qL

= +2m2hd

|Yd |2 + 2m2hu

|Yu|2 + m2qL

|Yd |2 + m2qL

|Yu|2 + 2Y †d m2

dRYd

+ |Yd |2m2qL

+ 2Y †u m2

uRYu + |Yu|2m2

qL+

1√15

g11σ1,1 (B.38)

β(2)m2

qL

= +85

g21m2

hu|Yu|2 − 4m2

hu|λu|2|Yu|2 − 2m2

Ru|λu|2|Yu|2

− 2m2S |λu|2|Yu|2 − 6m2

hu|Λu|2|Yu|2 − 3m2

Ru|Λu|2|Yu|2

− 3m2T |Λu|2|Yu|2 +

25

g21m2

qL|Yd |2 − |λd |2m2

qL|Yd |2

−32

|Λd |2m2qL

|Yd |2 +45

g21m2

qL|Yu|2 − |λu|2m2

qL|Yu|2

−32

|Λu|2m2qL

|Yu|2 +45

g21Y †

d m2dR

Yd − 2|λd |2Y †d m2

dRYd

− 3|Λd |2Y †d m2

dRYd +

25

g21 |Yd |2m2

qL− |λd |2|Yd |2m2

qL

−32

|Λd |2|Yd |2m2qL

+85

g21Y †

u m2uR

Yu − 2|λu|2Y †u m2

uRYu

− 3|Λu|2Y †u m2

uRYu +

45

g21 |Yu|2m2

qL− |λu|2|Yu|2m2

qL

−32

|Λu|2|Yu|2m2qL

− 8m2hd

Y †d |Yd |2Yd − 8m2

huY †

u |Yu|2Yu − 2m2qL

Y †d |Yd |2Yd

− 2m2qL

Y †u |Yu|2Yu − 4Y †

d m2dR

|Yd |2Yd − 4|Yd |2m2qL

|Yd |2 − 4Y †d |Yd |2m2

dRYd

− 2Y †d |Yd |2Ydm2

qL− 4Y †

u m2uR

|Yu|2Yu − 4|Yu|2m2qL

|Yu|2 − 4Y †u |Yu|2m2

uRYu

− 2Y †u |Yu|2Yum2

qL+

215

1(

45g42σ2,2 + 80g4

3σ2,3 + g1

(2√

15σ3,1 + g1σ2,11

))− 3m2

qL|Yd |2Tr

(|Yd |2

)− 6Y †

d m2dR

YdTr(

|Yd |2)

− 3|Yd |2m2qL

Tr(

|Yd |2)

− m2qL

|Yd |2Tr(

|Ye|2)

− 2Y †d m2

dRYdTr

(|Ye|2

)− |Yd |2m2

qLTr(

|Ye|2)

− 3|Yu|2m2qL

Tr(

|Yu|2)

− 12m2hu

|Yu|2Tr(

|Yu|2)

− 3m2qL

|Yu|2Tr(

|Yu|2)

− 6Y †u m2

uRYuTr

(|Yu|2

)+ |Yd |2

(45

g21m2

hd− 2(

2m2hd

+ m2Rd

+ m2S

)|λd |2 − 3

(2m2

hd+ m2

Rd+ m2

T

)|Λd |2

125

B. MRSSM renormalization group functions

− 12m2hd

Tr(

|Yd |2)

− 4m2hd

Tr(

|Ye|2)

− 6Tr(

m2dR

|Yd |2)

− 2Tr(

m2eR

|Ye|2)

− 2Tr(

m2˜L

|Ye|2)

− 6Tr(

m2qL

|Yd |2))

− 6|Yu|2Tr(

m2qL

|Yu|2)

− 6|Yu|2Tr(

m2uR

|Yu|2)

(B.39)

β(1)m2

˜L

= 2m2hd

|Ye|2 + 2Y †e m2

eRYe −

√35

g11σ1,1 + m2˜L

|Ye|2 + |Ye|2m2˜L

(B.40)

β(2)m2

˜L

= +65

g21m2

˜L|Ye|2 − |λd |2m2

˜L|Ye|2 −

32

|Λd |2m2˜L

|Ye|2 +125

g21Y †

e m2eR

Ye

− 2|λd |2Y †e m2

eRYe − 3|Λd |2Y †

e m2eR

Ye +65

g21 |Ye|2m2

˜L

− |λd |2|Ye|2m2˜L

−32

|Λd |2|Ye|2m2˜L

− 8m2hd

Y †e |Ye|2Ye − 2m2

˜LY †

e |Ye|2Ye

− 4Y †e m2

eR|Ye|2Ye − 4|Ye|2m2

˜L|Ye|2 − 4Y †

e |Ye|2m2eR

Ye − 2Y †e |Ye|2Yem2

˜L

+ 1(

6g42σ2,2 +

25

g1

(− 2√

15σ3,1 + 3g1σ2,11

))− 3m2

˜L|Ye|2Tr

(|Yd |2

)− 6Y †

e m2eR

YeTr(

|Yd |2)

− 3|Ye|2m2˜L

Tr(

|Yd |2)

− m2˜L

|Ye|2Tr(

|Ye|2)

− 2Y †e m2

eRYeTr

(|Ye|2

)− |Ye|2m2

˜LTr(

|Ye|2)

−15

|Ye|2(

10(

2m2hd

+ m2Rd

+ m2S

)|λd |2 + 15

(2m2

hd+ m2

Rd+ m2

T

)|Λd |2

+ 2(

− 6g21m2

hd+ 30m2

hdTr(

|Yd |2)

+ 10m2hd

Tr(

|Ye|2)

+ 15Tr(

m2dR

|Yd |2)

+ 5Tr(

m2eR

|Ye|2)

+ 5Tr(

m2˜L

|Ye|2)

+ 15Tr(

m2qL

|Yd |2)))

(B.41)

β(1)m2

hd

= +2(

m2hd

+ m2Rd

+ m2S

)|λd |2 + 3

(m2

hd+ m2

Rd+ m2

T

)|Λd |2 −

√35

g1σ1,1 + 6m2hd

Tr(

|Yd |2)

+ 2m2hd

Tr(

|Ye|2)

+ 6Tr(

m2dR

|Yd |2)

+ 2Tr(

m2eR

|Ye|2)

+ 2Tr(

m2˜L

|Ye|2)

+ 6Tr(

m2qL

|Yd |2)

(B.42)

β(2)m2

hd

= −12|λd |4(

m2hd

+ m2Rd

+ m2S

)− 15|Λd |4

(m2

hd+ m2

Rd+ m2

T

)− 2|λd |2

(2|λu|2

(2m2

S + m2hd

+ m2hu

+ m2Rd

+ m2Ru

)+ 3|Λd |2

(2m2

hd+ 2m2

Rd+ m2

S + m2T

))+ 3|Λd |2

(4g2

2

(m2

hd+ m2

Rd+ m2

T

)− |Λu|2

(2m2

T + m2hd

+ m2hu

+ m2Rd

+ m2Ru

))+

25

(15g4

2σ2,2 + 3g21σ2,11 − 2

√15g1σ3,1 − 2

(− 40g2

3 + g21

)m2

hdTr(

|Yd |2)

+ 6g21m2

hdTr(

|Ye|2)

− 2g21Tr(

m2dR

|Yd |2)

+ 80g23Tr(

m2dR

|Yd |2)

+ 6g21Tr(

m2eR

|Ye|2)

+ 6g21Tr(

m2˜L

|Ye|2)

− 2g21Tr(

m2qL

|Yd |2)

+ 80g23Tr(

m2qL

|Yd |2)

− 90m2hd

Tr(

|Yd |4)

− 15m2hd

Tr(

|Yu|2|Yd |2)

− 15m2hu

Tr(

|Yu|2|Yd |2)

− 30m2hd

Tr(

|Ye|4)

− 90Tr(

m2dR

|Yd |4)

− 15Tr(

m2dR

|Yu|2|Yd |2)

− 30Tr(

m2eR

|Ye|4)

− 30Tr(

m2˜L

Y †e |Ye|2Ye

)− 90Tr

(m2

qLY †

d |Yd |2Yd

)− 15Tr

(m2

qL|Yd |2|Yu|2

)− 15Tr

(m2

qL|Yu|2|Yd |2

)− 15Tr

(m2

uR|Yd |2|Yu|2

))(B.43)

β(1)m2

hu= +2

(m2

hu+ m2

Ru+ m2

S

)|λu|2 + 3

(m2

hu+ m2

Ru+ m2

T

)|Λu|2 +

√35

g1σ1,1 + 6m2hu

Tr(

|Yu|2)

+ 6Tr(

m2qL

|Yu|2)

+ 6Tr(

m2uR

|Yu|2)

(B.44)

126

B.3. Beta functions of soft breaking parameters

β(2)m2

hu= +12g2

2m2hu

|Λu|2 + 12g22m2

Ru|Λu|2 + 12g2

2m2T |Λu|2

− 4|λu|2|λd |2(

2m2S + m2

hd+ m2

hu+ m2

Rd+ m2

Ru

)− 12|λu|4

(m2

hu+ m2

Ru+ m2

S

)− 6|Λu|2λu|2

(2m2

hu+ 2m2

Ru+ m2

S + m2T

)| − 3|Λu|2|Λd |2

(m2

hd+ m2

hu

)− 3|Λu|2|Λd |2

(m2

Rd+ m2

Ru|)

− 6|Λu|2m2T |Λd |2

− 15|Λ4u|(

m2Ru

+ m2T + m2

hu

)+ 6g4

2σ2,2 +65

g21σ2,11 + 4

√35

g1σ3,1 +85

g21m2

huTr(

|Yu|2)

+ 32g23m2

huTr(

|Yu|2)

+85

g21Tr(

m2qL

|Yu|2)

+ 32g23Tr(

m2qL

|Yu|2)

+85

g21Tr(

m2uR

|Yu|2)

+ 32g23Tr(

m2uR

|Yu|2)

− 6m2hd

Tr(

|Yu|2|Yd |2)

− 6m2hu

Tr(

|Yu|2|Yd |2)

− 36m2hu

Tr(

|Yu|4)

− 6Tr(

m2dR

|Yu|2|Yd |2)

− 6Tr(

m2qL

|Yd |2|Yu|2)

− 6Tr(

m2qL

|Yu|2|Yd |2)

− 36Tr(

m2qL

Y †u |Yu|2Yu

)− 6Tr

(m2

uR|Yd |2|Yu|2

)− 36Tr

(m2

uR|Yu|4

)(B.45)

β(1)m2

dR

= 2(

2m2hd

|Yd |2 + 2Ydm2qL

Y †d + m2

dR|Yd |2 + |Yd |2m2

dR

)+ 2

1√15

g11σ1,1 (B.46)

β(2)m2

dR

= +25

g21m2

dR|Yd |2 + 6g2

2m2dR

|Yd |2 − 2|λd |2m2dR

|Yd |2 − 3|Λd |2m2dR

|Yd |2

+45

g21Ydm2

qLY †

d + 12g22Ydm2

qLY †

d − 4|λd |2Ydm2qL

Y †d − 6|Λd |2Ydm2

qLY †

d

+25

g21 |Yd |2m2

dR+ 6g2

2 |Yd |2m2dR

− 2|λd |2|Yd |2m2dR

− 3|Λd |2|Yd |2m2dR

− 8m2hd

|Yd |4 − 4m2hd

|Yu|2|Yd |2 − 4m2hu

|Yu|2|Yd |2

− 2m2dR

|Yd |4 − 2m2dR

|Yu|2|Yd |2 − 4Ydm2qL

Y †d |Yd |2 − 4Ydm2

qL|Yu|2Y †

d

− 4|Yd |2m2dR

|Yd |2 − 4|Yd |2Ydm2qL

Y †d − 2|Yd |4m2

dR− 4YdY †

u m2uR

YuY †d

− 4Yd |Yu|2m2qL

Y †d − 2|Yu|2|Yd |2m2

dR+

815

1(

20g43σ2,3 + g1

(g1σ2,11 +

√15σ3,1

))− 6m2

dR|Yd |2Tr

(|Yd |2

)− 12Ydm2

qLY †

d Tr(

|Yd |2)

− 6|Yd |2m2dR

Tr(

|Yd |2)

− 2m2dR

|Yd |2Tr(

|Ye|2)

− 4Ydm2qL

Y †d Tr(

|Ye|2)

− 2|Yd |2m2dR

Tr(

|Ye|2)

−25

|Yd |2(

− 2g21m2

hd− 30g2

2m2hd

+ 10(

2m2hd

+ m2Rd

+ m2S

)|λd |2

+ 15(

2m2hd

+ m2Rd

+ m2T

)|Λd |2 + 60m2

hdTr(

|Yd |2)

+ 20m2hd

Tr(

|Ye|2)

+ 30Tr(

m2dR

|Yd |2)

+ 10Tr(

m2eR

|Ye|2)

+ 10Tr(

m2˜L

|Ye|2)

+ 30Tr(

m2qL

|Yd |2))

(B.47)

β(1)m2

uR

= 2(

2m2hu

|Yu|2 + 2Yum2qL

Y †u + m2

uR|Yu|2 + |Yu|2m2

uR

)− 4

1√15

g11σ1,1 (B.48)

β(2)m2

uR

= −25

g21m2

uR|Yu|2 + 6g2

2m2uR

|Yu|2 − 2|λu|2m2uR

|Yu|2 − 3|Λu|2m2uR

|Yu|2

−45

g21Yum2

qLY †

u + 12g22Yum2

qLY †

u − 4|λu|2Yum2qL

Y †u − 6|Λu|2Yum2

qLY †

u

−25

g21 |Yu|2m2

uR+ 6g2

2 |Yu|2m2uR

− 2|λu|2|Yu|2m2uR

− 3|Λu|2|Yu|2m2uR

− 4m2hd

|Yd |2|Yu|2 − 4m2hu

|Yd |2|Yu|2 − 8m2hu

|Yu|4

127

B. MRSSM renormalization group functions

− 2m2uR

|Yd |2|Yu|2 − 2m2uR

|Yu|4 − 4Yum2qL

|Yd |2Y †u − 4Yum2

qLY †

u |Yu|2

− 4YuY †d m2

dRYdY †

u − 4Yu|Yd |2m2qL

Y †u − 2|Yd |2|Yu|2m2

uR

− 4|Yu|2m2uR

|Yu|2 − 4|Yu|2Yum2qL

Y †u − 2|Yu|4m2

uR

+1615

1(

10g43σ2,3 + g1

(2g1σ2,11 −

√15σ3,1

))− 6m2

uR|Yu|2Tr

(|Yu|2

)− 12Yum2

qLY †

u Tr(

|Yu|2)

− 6|Yu|2m2uR

Tr(

|Yu|2)

−25

|Yu|2(

2g21m2

hu− 30g2

2m2hu

+ 10(

2m2hu

+ m2Ru

+ m2S

)|λu|2 + 15

(2m2

hu+ m2

Ru+ m2

T

)|Λu|2

+ 60m2hu

Tr(

|Yu|2)

+ 30Tr(

m2qL

|Yu|2)

+ 30Tr(

m2uR

|Yu|2))

(B.49)

β(1)m2

eR

= 2(

2m2hd

|Ye|2 + 2Yem2˜L

Y †e + m2

eR|Ye|2 + |Ye|2m2

eR

)+ 2

√35

g11σ1,1 (B.50)

β(2)m2

eR

= −65

g21m2

eR|Ye|2 + 6g2

2m2eR

|Ye|2 − 2|λd |2m2eR

|Ye|2 − 3|Λd |2m2eR

|Ye|2

−125

g21Yem2

˜LY †

e + 12g22Yem2

˜LY †

e − 4|λd |2Yem2˜L

Y †e − 6|Λd |2Yem2

˜LY †

e

−65

g21 |Ye|2m2

eR+ 6g2

2 |Ye|2m2eR

− 2|λd |2|Ye|2m2eR

− 3|Λd |2|Ye|2m2eR

− 8m2hd

|Ye|4 − 2m2eR

|Ye|4 − 4Yem2˜L

Y †e |Ye|2 − 4|Ye|2m2

eR|Ye|2

− 4|Ye|2Yem2˜L

Y †e − 2|Ye|4m2

eR+

85

g11(

3g1σ2,11 +√

15σ3,1

)− 6m2

eR|Ye|2Tr

(|Yd |2

)− 12Yem2

˜LY †

e Tr(

|Yd |2)

− 6|Ye|2m2eR

Tr(

|Yd |2)

− 2m2eR

|Ye|2Tr(

|Ye|2)

− 4Yem2˜L

Y †e Tr(

|Ye|2)

− 2|Ye|2m2eR

Tr(

|Ye|2)

−25

|Ye|2(

10(

2m2hd

+ m2Rd

+ m2S

)|λd |2 + 15

(2m2

hd+ m2

Rd+ m2

T

)|Λd |2

+ 2(

3g21m2

hd− 15g2

2m2hd

+ 30m2hd

Tr(

|Yd |2)

+ 10m2hd

Tr(

|Ye|2)

+ 15Tr(

m2dR

|Yd |2)

+ 5Tr(

m2eR

|Ye|2)

+ 5Tr(

m2˜L

|Ye|2)

+ 15Tr(

m2qL

|Yd |2)))

(B.51)

β(1)m2

S= 4((

m2hd

+ m2Rd

+ m2S

)|λd |2 +

(m2

hu+ m2

Ru+ m2

S

)|λu|2

)(B.52)

β(2)m2

S= −

45

(20|λd |4

(m2

hd+ m2

Rd+ m2

S

)− 15|Λu|2

(2m2

hu+ 2m2

Ru+ m2

S + m2T

)+ |λd |2

(− 3g2

1m2hd

− 15g22m2

hd− 3g2

1m2Rd

− 15g22m2

Rd− 3g2

1m2S − 15g2

2m2S

+ 15|Λd |2(

2m2hd

+ 2m2Rd

+ m2S + m2

T

)+ 15

(2m2

hd+ m2

Rd+ m2

S

)Tr(

|Yd |2)

+ 10m2hd

Tr(

|Ye|2)

+ 5m2Rd

Tr(

|Ye|2)

+ 5m2STr(

|Ye|2)

+ 15Tr(

m2dR

|Yd |2)

+ 5Tr(

m2eR

|Ye|2)

+ 5Tr(

m2˜L

|Ye|2)

+ 15Tr(

m2qL

|Yd |2))

+ |λu|2(

20|λu|2(

m2hu

+ m2Ru

+ m2S

)− 3(

g21m2

hu+ 5g2

2m2hu

+ g21m2

Ru+ 5g2

2m2Ru

+ g21m2

S + 5g22m2

S

− 5(

2m2hu

+ m2Ru

+ m2S

)Tr(

|Yu|2)

− 5Tr(

m2qL

|Yu|2)

− 5Tr(

m2uR

|Yu|2))))

(B.53)

β(1)m2

T= 2((

m2hd

+ m2Rd

+ m2T

)|Λd |2 +

(m2

hu+ m2

Ru+ m2

T

)|Λu|2

)(B.54)

β(2)m2

T= −

25

(30|Λd |4

(m2

hd+ m2

Rd+ m2

T

)30|Λu|4

(m2

hu+ m2

Ru+ m2

T

)− 40g4

2σ2,2

+ |Λd |2(

− 3g21m2

hd+ 5g2

2m2hd

− 3g21m2

Rd+ 5g2

2m2Rd

− 3g21m2

T + 5g22m2

T

128

B.3. Beta functions of soft breaking parameters

+ 10|λd |2(

2m2hd

+ 2m2Rd

+ m2S + m2

T

)+ 15

(2m2

hd+ m2

Rd+ m2

T

)Tr(

|Yd |2)

+ 10m2hd

Tr(

|Ye|2)

+ 5m2Rd

Tr(

|Ye|2)

+ 5m2T Tr(

|Ye|2)

+ 15Tr(

m2dR

|Yd |2)

+ 5Tr(

m2eR

|Ye|2)

+ 5Tr(

m2˜L

|Ye|2)

+ 15Tr(

m2qL

|Yd |2))

+ |Λu|2(

− 3g21m2

hu+ 5g2

2m2hu

− 3g21m2

Ru+ 5g2

2m2Ru

− 3g21m2

T + 5g22m2

T

+ 10|λu|2(

2m2hu

+ 2m2Ru

+ m2S + m2

T

)+ 15

(2m2

hu+ m2

Ru+ m2

T

)Tr(

|Yu|2)

+ 15Tr(

m2qL

|Yu|2)

+ 15Tr(

m2uR

|Yu|2)))

(B.55)

β(1)m2

O= 0 (B.56)

β(2)m2

O= 24g4

3σ2,3 (B.57)

β(1)m2

Rd

= 2(

m2hd

+ m2Rd

+ m2S

)|λd |2 + 3

(m2

hd+ m2

Rd+ m2

T

)|Λd |2 +

√35

g1σ1,1 (B.58)

β(2)m2

Rd

= −12|λd |4(

m2hd

+ m2Rd

+ m2S

)− 15|Λd |4

(m2

hd+ m2

Rd+ m2

T

)+ 6g4

2σ2,2 +65

g21σ2,11

+ 4

√35

g1σ3,1 + 3|Λd |2(

4g22

(m2

hd+ m2

Rd+ m2

T

)− |Λu|2

(2m2

T + m2hd

+ m2hu

+ m2Rd

+ m2Ru

)− 3(

2m2hd

+ m2Rd

+ m2T

)Tr(

|Yd |2)

− 2m2hd

Tr(

|Ye|2)

− m2Rd

Tr(

|Ye|2)

− m2T Tr(

|Ye|2)

− 3Tr(

m2dR

|Yd |2)

− Tr(

m2eR

|Ye|2)

− Tr(

m2˜L

|Ye|2)

− 3Tr(

m2qL

|Yd |2))

− 2|λd |2(

2|λu|2(

2m2S + m2

hd+ m2

hu+ m2

Rd+ m2

Ru

)+ 3|Λd |2

(2m2

hd+ 2m2

Rd+ m2

S + m2T

)+ 6m2

hdTr(

|Yd |2)

+ 3m2Rd

Tr(

|Yd |2)

+ 3m2STr(

|Yd |2)

+ 2m2hd

Tr(

|Ye|2)

+ m2Rd

Tr(

|Ye|2)

+ m2STr(

|Ye|2)

+ 3Tr(

m2dR

|Yd |2)

+ Tr(

m2eR

|Ye|2)

+ Tr(

m2˜L

|Ye|2)

+ 3Tr(

m2qL

|Yd |2))(B.59)

β(1)m2

Ru= 2(

m2hu

+ m2Ru

+ m2S

)|λu|2 + 3

(m2

hu+ m2

Ru+ m2

T

)|Λu|2 −

√35

g1σ1,1 (B.60)

β(2)m2

Ru= +12g2

2m2hu

|Λu|2 + 12g22m2

Ru|Λu|2 + 12g2

2m2T |Λu|2

− 4|λu|2|λd |2(

2m2S + m2

hd+ m2

hu+ m2

Rd+ m2

Ru

)− 12|λu|4

(m2

hu+ m2

Ru+ m2

S

)− 3m2

hd|Λd |2|Λu|2 − 3|Λu|2|Λd |2

(m2

hu+ m2

Rd+ Λum2

Ru

)− 6|Λu|2|Λd |2m2

T

− 15|Λu|4(

m2hu

+ m2Ru

+ m2T

)+ 6g4

2σ2,2 +65

g21σ2,11 − 4

√35

g1σ3,1

− 18m2hu

|Λu|2Tr(

|Yu|2)

− 9m2Ru

|Λu|2Tr(

|Yu|2)

− 9m2T |Λu|2Tr

(|Yu|2

)− 9|Λu|2Tr

(m2

qL|Yu|2

)− 9|Λu|2Tr

(m2

uR|Yu|2

)− 6|λu|2

((2m2

hu+ m2

Ru+ m2

S

)Tr(

|Yu|2)

+ |Λu|2(

2m2hu

+ 2m2Ru

+ m2S + m2

T

)+ Tr

(m2

qL|Yu|2

)+ Tr

(m2

uR|Yu|2

))(B.61)

129

C. THE MRSSM AFTERELECTROWEAK SYMMETRYBREAKING

After EWSB the singlet and triplet vacuum expectation values effectively modify theµ-parameters of the model, and it is useful to define the abbreviations

µeff,±i = µi +

λivS√2± ΛivT

2, µeff,0

i = µi +λivS√

2, i = u, d. (C.1)

C.1. SCALAR POTENTIAL AND TADPOLE EQUATIONS

The electroweak part of the tree-level scalar potential for the neutral fields takes the formof

V EW =(

m2hd

+ µ2d

) ∣∣∣h0d

∣∣∣2 +(

m2hu

+ µ2u

) ∣∣∣h0u

∣∣∣2 − Bµ(

h0dh0

u + h.c.)

(C.2)

+(

m2Rd

+ µ2d

) ∣∣∣R0d

∣∣∣2 +(

m2Ru

+ µ2u

) ∣∣∣R0u

∣∣∣2 + m2T

∣∣∣T 0∣∣∣2 + m2

S |S|2

+(

2MDW<

(T 0))2

−(

2MDB<(S)

)2+ 1

8 (g21 + g2

2 )(∣∣h0

d

∣∣2 −∣∣h0

u

∣∣2 −∣∣R0

d

∣∣2 +∣∣R0

u

∣∣2)2

−[(

g1MDB −√

2λdµd

)√2<(S) −

(g2MD

W + Λdµd

)√2<(

T 0)

−∣∣∣λdS + 1√

2ΛdT 0

∣∣∣2] ∣∣∣h0d

∣∣∣2+[(

g1MDB −√

2λuµu

)√2<(S) −

(g2MD

W + Λuµu

)√2<(

T 0)

+∣∣∣λuS − 1√

2ΛuT 0

∣∣∣2] ∣∣∣h0u

∣∣∣2+[(

g1MDB +√

2λdµd

)√2<(S) −

(g2MD

W − Λdµd

)√2<(

T 0)

+∣∣∣λdS + 1√

2ΛdT 0

∣∣∣2] ∣∣∣R0d

∣∣∣2−[(

g1MDB −√

2λuµu

)√2<(S) −

(g2MD

W − Λuµu

)√2<(

T 0)

−∣∣∣λuS − 1√

2ΛuT 0

∣∣∣2] ∣∣∣R0u

∣∣∣2+(λ2

d + 12Λd

) ∣∣∣h0dR0

d

∣∣∣2 +(λ2

u + 12Λ

2u

) ∣∣∣h0uR0

u

∣∣∣2 −(λdλu − 1

2ΛdΛu) (

h0dR0

dh0*u R0*

u + h.c.)

The minimization conditions for the tree-level scalar potential are given as

0 = td = tu = tT = tS , (C.3)

131

C. The MRSSM after electroweak symmetry breaking

where

td =vd [18

(g2

1 + g22

) (v2

d − v2u)

− g1MDB vS + g2MD

W vT + m2hd

+ (µeff,+d )2] − vuBµ ,

tu =vu[18

(g2

1 + g22

) (v2

u − v2d

)+ g1MD

B vS − g2MDW vT + m2

hu+ (µeff,−

u )2] − vdBµ ,

tT =12g2MD

W

(v2

d − v2u)

+ 12Λdv2

dµeff,+d − Λuv2

uµeff,−u + 4(MD

W )2vT + m2T vT ,

tS =12g1MD

B

(v2

u − v2d

)+ 1√

2λdv2

dµeff,+d + λuv2

uµeff,−u + 4(MD

B )2vS + m2SvS . (C.4)

C.2. MASSES OF THE HIGGS FIELDS

Using the tadpole eqs. (C.4), the pseudo-scalar Higgs boson mass matrix in the basis(σd ,σu,σS,σT ) has a simple form (in Landau gauge)

MA =

Bµ vu

vdBµ 0 0

Bµ Bµ vdvu

0 0

0 0 m2S + λ2

d v2d +λ2

uv2u

2λdΛd v2

d −λuΛuv2u

2√

2

0 0 λdΛd v2d −λuΛuv2

u

2√

2m2

T + Λ2d v2

d +Λ2uv2

u4

. (C.5)

The scalar Higgs boson mass matrix in the weak basis (φd ,φu,φS,φT ) is given by

MH0 =

(MMSSM MT

21M21 M22

)(C.6)

with the sub-matrices (cβ = cosβ, sβ = sinβ, etc)

MMSSM =

(m2

Z c2β + m2

As2β −(m2

Z + m2A)sβcβ

−(m2Z + m2

A)sβcβ m2Z s2

β + m2Ac2

β

),

M22 =

4(MDB )2 + m2

S + λ2d v2

d +λ2uv2

u2

λdΛd v2d −λuΛuv2

u

2√

2λdΛd v2

d −λuΛuv2u

2√

24(MD

W )2 + m2T + Λ2

d v2d +Λ2

uv2u

4

,

M21 =

(vd (√

2λdµeff,+d − g1MD

B ) vu(√

2λuµeff,−u + g1MD

B )vd (Λdµ

eff,+d + g2MD

W ) −vu(Λuµeff,−u + g2MD

W )

).

The charged Higgs boson mass matrix in the weak basis (H−*d , H+

u , T −*, T +) is given as

MH± =

(MMSSM,± MT

21,±M21,± M22,±

)(C.7)

with the sub-matrices

MMSSM,± =

(m2

H±c2β − 2vT (g2MD

W + Λdµeff,0d ) m2

H±sβcβm2

H±sβcβ m2H±c2

β + 2vT (g2MDW + Λuµ

eff,0u )

),

M22,± =

(2(MD

W )2 + m2T 2(MD

W )2

2(MDW )2 2(MD

W )2 + m2T

)(C.8)

+

g22v2

T +Λ2d v2c2

β

2 − g22v2 cos 2β

4 −g22v2

T2

−g22v2

T2

g22v2

T +Λ2uv2s2

β

2 + g22v2 cos 2β

4

,

132

C.3. Mass matrices of R-Higgses

M21,± =

(vd

2√

2(2Λdµ

eff,−d + 2g2MD

W + vT g22 ) vu

2√

2(2Λuµ

eff,−u + 2g2MD

W + vT g22 )

vd

2√

2(2Λdµ

eff,+d + 2g2MD

W − vT g22 ) vu

2√

2(2Λuµ

eff,+u + 2g2MD

W − vT g22 )

),

C.3. MASS MATRICES OF R-HIGGSES

The mass matrix of neutral R-Higgs bosons is

MR =

(m2

Rd Rd

14

(ΛuΛd − 2λuλd

)vuvd

14

(ΛuΛd − 2λuλd

)vuvd m2

RuRu

)(C.9)

with

m2Rd Rd

= m2Rd

+ (µeff,+d )2 + g1MD

B vS − g2MDW vT + 1

8 [(g21 + g2

2 )(v2u − v2

d ) + 4λ2dv2

d + 2Λ2dv2

d ],

m2RuRu

= m2Ru

+ (µeff,−u )2 − g1MD

B vS + g2MDW vT + 1

8 [(g21 + g2

2 )(v2d − v2

u ) + 4λ2uv2

u + 2Λ2uv2

u ].

It is diagonalized by an orthogonal matrix with mixing angle θR.

Because of the R-symmetry the charged R-Higgs bosons do not mix and the masseigenvalues are given as

m2R+

1= m2

Rd+ (µeff,−

d )2 + g1MDB vS + g2MD

W vT + 18 [(g2

1 − g22 )(v2

u − v2d ) + 4Λ2

dv2d ]

m2R+

2= m2

Ru+ (µeff,+

u )2 − g1MDB vS − g2MD

W vT + 18 [(g2

1 − g22 )(v2

d − v2u ) + 4Λ2

uv2u ] . (C.10)

C.4. MASS MATRICES OF SQUARKS AND SLEPTONS

Due to their different R-charges, left- and right-handed sfermions do not mix and the sectorscan be diagonalised independently so that only 3 × 3 mixing and not the most general 6 × 6mixing has to be considered.

SQUARKS

The down squarks have the two sets of gauge eigenstate basis d*iL, djL and d*

iR, djR withi , j ∈ 1, 2, 3. Their respective mass matrices are given as

m2dLd*

L= m2

q + |Yd |2v2

d

2+ 13

(g2

1

24

(v2

u − v2d

)+

g22

8

(v2

u − v2d

)+

g1MDB vS

3− g2MD

W vT

)(C.11)

m2dR d*

R= m2

dR+ |Yd |2

v2d

2+ 13

(g2

1

12

(v2

u − v2d

)+

2g1MDB vS

3

). (C.12)

These matrices are diagonalized by ZDL and ZD

R , respectively:

ZD,†L/ Rm2

dL/ R d*L/ R

ZDL/ R = m2,diag

d,L/ R(C.13)

133

C. The MRSSM after electroweak symmetry breaking

with the squark mass eigenstates defined as

ddiagL,i =

∑j

ZDL,ij dL,j , ddiag

R,i =∑

j

ZDR,ij dR,j . (C.14)

The up squarks have the two sets of gauge eigenstate basis uiL and ujR with i , j ∈ 1, 2, 3.Their respective mass matrices are given as

m2uLu*

L= m2

q + |Yu|2v2

u

2+ 13

(g2

1

24

(v2

u − v2d

)−

g22

8

(v2

u − v2d

)+

g1MDB vS

3+ g2MD

W vT

)(C.15)

m2uR u*

R= m2

uR+ |Yu|2

v2u

2+ 13

(g2

1

6

(v2

d − v2u

)−

4g1MDB vS

3

). (C.16)

These matrices are diagonalized by ZDL and ZD

R , respectively:

ZUL/ Rm2

uL/ R u*L/ R

ZU,†L/ R = m2,diag

u,L/ R (C.17)

with the squark mass eigenstates defined as

udiagL,i =

∑j

ZUL,ij ujL , udiag

R,i =∑

j

ZUR,ij ujR . (C.18)

SLEPTONS

The sneutrinos have the gauge eigenstate basis νiL with i ∈ 1, 2, 3. The mass matrix isgiven as

m2νLν* = m2

˜ + 13

(g2

1

8

(v2

d − v2u

)+

g22

8

(v2

d − v2u

)− g1MD

B vS − g2MDW vT

), (C.19)

which is diagonalised by ZV :ZV,†m2

νν*ZV = m2,diagν (C.20)

with the sneutrino mass eigenstates defined as

νdiagi =

∑j

ZVij νLj . (C.21)

The charged sleptons have the two sets of gauge eigenstate basis eiL and ejR withi , j ∈ 1, 2, 3. Their respective mass matrices are given as

m2eLe*

L= m2

˜ + |Ye|2v2

d

2+ 13

(g2

1

8

(v2

d − v2u

)+

g22

8

(v2

u − v2d

)− g1MD

B vS − g2MDW vT

)(C.22)

m2eR e*

R= m2

eR+ |Ye|2

v2d

2+ 13

(g2

1

4

(v2

u − v2d

)+ 2g1MD

B vS

). (C.23)

These matrices are diagonalized by ZEL and ZE

R , respectively:

ZE,†L/ Rm2

eL/ R e*L/ R

ZEL/ R = m2,diag

e,L/ R (C.24)

134

C.5. Mass matrices of neutralinos and charginos

with the squark mass eigenstates defined as

ediagL,i =

∑j

ZEL,ij ejL , ediag

R,i =∑

j

ZER,ij ejR . (C.25)

C.5. MASS MATRICES OF NEUTRALINOS AND CHARGINOS

In the weak basis of eight neutral electroweak two-component fermions: ξi = (B, W 0, R0d , R0

u),ζi = (S, T 0, H0

d , H0u ) with R-charges +1, −1 respectively, the neutralino mass matrix takes

the form of

mχ =

MD

B 0 −12g1vd

12g1vu

0 MDW

12g2vd −1

2g2vu

− 1√2λdvd −1

2Λdvd −µeff,+d 0

1√2λuvu −1

2Λuvu 0 µeff,−u

. (C.26)

The transformation to a diagonal mass matrix and mass eigenstates κi and ψi is performedby two unitary mixing matrices N1 and N2 as

N1,*mχN2,† = mdiagχ , ξi =

4∑j=1

N1,*j i κj , ζi =

4∑j=1

N2,*i j ψj ,

and physical four-component Dirac neutralinos are constructed as

χi =

(κi

ψ*i

)i = 1, 2, 3, 4. (C.27)

The mass matrix of charginos in the weak basis of eight charged two-component fermionsbreaks into two (2 × 2) submatrices. The first, in the basis ξ+

i = (W +, R+d ) and ζ−

i = (T −, H−d )

of spinors with R-charge equal to electric charge, takes the form of

mχ+ =

(g2vT + MD

W1√2Λdvd

1√2g2vd +µeff,−

d

)(C.28)

The diagonalization and transformation to mass eigenstates λ±i is performed by two unitarymatrices U1 and V 1 as

U1,*mχ+V 1,† = mdiagχ+ , ζ−

i =2∑

j=1

U1,*j i λ

−j ξ+

i =2∑

j=1

V 1,*i j λ+

j (C.29)

and the corresponding physical four-component charginos are built as

χ+i =

(λ+

iλ−*

i

)i = 1, 2. (C.30)

The second submatrix, in the basis ξ−i = (W −, R−

u ), ζ+i = (T +, H+

u ) of spinors with R-chargeequal to minus electric charge, reads

mρ− =

(−g2vT + MD

W1√2g2vu

− 1√2Λuvu −µeff,+

u

)(C.31)

135

C. The MRSSM after electroweak symmetry breaking

The diagonalization and transformation to mass eigenstates η±i is performed by U2 and V 2

as

U2,*mρ−V 2,† = mdiagρ− , ξ−

i =2∑

j=1

U2,*j i η

−j ζ+

i =2∑

j=1

V 2,*i j η+

j (C.32)

and the other two physical four-component charginos are built as

ρ−i =

(η−

iη+*

i

)i = 1, 2. (C.33)

The mixing between the different gauge eigenstates to the charginos and neutralinosis normally not large so that the mass eigenstates are almost pure gauge eigenstates. Anaming convention following the MSSM is possible but should keep the Dirac nature of theneutralinos in mind. Hence, depending on their predominant component also the masseigenstates are called bino-singlino, wino-triplino and up- and down-higgsino in this work.

136

D. EXOTIC DECAY PATTERNS OFSTATES FOR THE LIGHT SINGLETSCENARIO

With the requirement of a light scalar boson with a mass below 125 GeV it is possible thatone or several of the new light states could become decay products of Standard Modelfields, especially the 125 GeV Higgs and Z boson. In the following, the possible decayrates to those fields are discussed and it is shown that for the realistic parameter spacein agreement with Higgs observables in section 4.4 (and also dark matter as shown inchapter 6) no problems or further constraints arise. In the following, it is assumed thatλu, λd , the doublet mixing into the lightest Higgs boson ZH

11, ZH12 and the non-singlino-bino

components in the LSP N i12, N i

13, N i14 are small and of similar magnitude, so that only

contributions with the lowest power in those parameters are dominant and need to bediscussed here.

Generally, for all decays considered, the decay rate into the new light particles aresuppressed in scenarios obeying LHC and dark matter bounds, as λu, λd are required to bevery small by the hierarchy (4.13). These parameters are responsible for the singlet-doublet(singlino-higgsino) mixing and together with the requirements from HiggsBounds andHiggsSignals in the mass region of interest they put very stringent limits on the mixingmatrix elements ZH

11, ZH12. This can be also seen from figure 4.7, where (ZH

12)2 < 0.02 fora considerable part of the parameter space. For the LSP, the rather large µu, µd requiredby dark matter direct detection and the smallness of λu,d constrain the neutralino mixingmatrix elements N i

13, N i14. Emphasis is put on the decay into a pair of LSP as this final state

contributes to the branching fraction into invisible states. The decay width for a two-bodydecay of a field A into fields D1 and D2 is given as

ΓA−>D1D2 =S

8πmA

√√√√14

−m2

D1+ m2

D2

2m2A

|MA−>D1D2 |2 (D.1)

with the matrix element |M|2 and a statistical factor S = 1/ 2 (S = 1) if the final stateparticles are identical (not identical). The respective matrix elements are given the following.

137

D. Exotic decay patterns of states for the light singlet scenario

SM-LIKE HIGGS BOSON

Decay into light CP-odd state The decay of the SM-like Higgs to the singlet-like pseu-doscalar is possible for masses below mA1 ≈ mS < 125 GeV/ 2. The matrix element for thedecay H2 → A1A1 is

|MH2→A1A1 |2 = v2(λ2

d sinβ sinα − λ2u cosβ cosα

)2, (D.2)

where α is the usual CP-even Higgs mixing angle diagonalizing the sub-matrix of the up-and down-Higgs doublets. When using this angle in the MRSSM it is assumed that themixing with the singlet and triplet is negligible compared to the doublet mixing. In the largemA limit of the MSSM the matrix element reduces to

|MH2→A1A1 |2α=β−π/ 2

= v2(λ2

d cos2 β + λ2u sin2 β

)2. (D.3)

The size of contribution in realistic scenarios is negligible as λ is limited very strongly asdiscussed before.

Decay into light CP-even state The matrix element of the decay of the SM-like Higgsto a pair of singlet-like scalars is a more complicated as, unlike in the CP-odd case, mixingof the singlet and doublet is possible. It is given as

|MH2→H1H1 |2 =v2[(λ2

u sinβ cosα − λ2d cosβ sinα

)−

2Mφ13

v2 sinβZH

11 sinα +2Mφ

23

v2 cosβZH

12 cosα +m2

Z

v2

(2 cos(α + β)ZH

11ZH12

)+

m2Z

v2

(sin(α + β)

((ZH

11)2 − (ZH12)2

)− sin(α − β)

((ZH

11)2 + (ZH12)2

))]2

, (D.4)

where Mφ is the scalar Higgs mass matrix and ZH the scalar mixing matrix. The first linedescribes the contribution given by the decay of the doublet components in the heavierHiggs boson to the singlet component of the light one and is comparable to the expressionfor the CP-odd Higgs as decay product. The first two terms of the second line originatefrom the contributions of a doublet component of the heavier Higgs decaying to a doubletand a singlet component of the lighter Higgs. The last term of this line and the last linestem completely from the doublet components. As described for the CP-odd case, λcontributions are negligible. The other contributions can be estimated to be of similarsize, as for the relevant mass region of mH1 < 60 GeV HiggsBounds limits the doubletcomponents of the lightest state ((ZH

11)2, (ZH12)2) to at most two percent as given by fig. 4.7;

the off-diagonal scalar mass matrix elements Mφ13, Mφ

23 are suppressed compared to v2

by a similar magnitude.

Decay into LSP For the decay of the SM Higgs boson into a pair of LSPs the matrixelement is given as

|MH2→χ1χ1 |2 =m2

H1− 4m2

χ1

2

(cosα((

√2λuN2

11 − ΛuN212)N1

14 + (g1N111 − g2N1

12)N214)

138

+ sinα((√

2λdN211 + ΛdN2

12)N113 + (g1N1

11 − g2N112)N2

13))2

. (D.5)

This can be compared to the MSSM matrix element∣∣∣MMSSMH2→χ1χ1

∣∣∣2 = 2(m2H1

− 4m2χ1

)(g1N11 − g2N12)2(N13 sinα + N14 cosα)2 . (D.6)

This matrix element contributes to the invisible decay of the Higgs boson, but as thehiggsino content of the LSP N i

13 and N i14 necessarily enters, it is too small be experimentally

significant. The experimental upper limit for the branching ratio of invisible decay of the SMHiggs boson is at the moment at BRup = 0.28 at 95% C.L. [289]. The branching ratios forthe characteristic benchmark points are BRinv = 4 · 10−7, 4 · 10−6, 5 · 10−6 for BMP4, BMP5,BMP6, respectively.

DECAY OF THE Z BOSON INTO LSP

For completeness also the decay of Z boson to light MRSSM states is included. The Zboson does not couple to a singlet scalar or pseudo-scalar, hence only the decay intothe LSP needs to be discussed here. The decay width of the Z boson was studied veryprecisely at LEP, hence it needs to be ensured that the MRSSM satisfies the experimentalconstraints. The invisible partial width was measured to be Γinv = 499 ± 1.5 MeV [98].Therefore, the partial width to LSPs needs be below one MeV to be compatible withexperiment.

The Z boson couples to the LSP only through the (R-)higgsino components. The matrixelement of the decay is given as∣∣∣MZ→χ0

1χ01

∣∣∣2 =g2

2

2 cos2 θWm2

Z (Z21,3−4 + Z2

2,3−4) − m2χ1

(Z21,3−4 + Z2

2,3−4 − 6Z1,3−4Z2,3−4) , (D.7)

where Zi ,3−4 = (N i13)2 − (N i

14)2 is the difference of the mixing matrix elements squared forthe mixing of the bino with the down- and up-R-higgsino (singlino with the down- andup-higgsino) when i = 1 (i = 2). If the mixing is similar, e.g. Z1,3−4 = Z2,3−4 = Z3−4, thissimplifies to ∣∣∣MZ→χ0

1χ01

∣∣∣2 Z1,3−4≈Z2,3−4=g2

2

cos2 θW(m2

Z + 2m2χ1

)Z23−4 . (D.8)

This can be compared to the MSSM result∣∣∣MMSSMZ→χ0

1χ01

∣∣∣2 =g2

2

cos2 θW(m2

Z − 4m2χ0

1)Z2

3−4,MSSM , (D.9)

where Z3−4,MSSM = (N13)2 − (N14)2 is similarly defined as the difference of squared matrixelements of the higgsino mixing in the LSP. The difference to eq. (D.8) comes from theDirac nature of the LSP in the MRSSM. It has to be noted as stated before that the structureof the neutralino mixing matrices in the usual MSSM and MRSSM are quite different asthe µ terms of the MRSSM appear on the main diagonal of the mass matrix.

For illustration the limit tanβ 1, λu = 0, and µu MDB , g1v can be used for the

neutralino mass matrix to deduce a simplified expression for Zi ,3−4 + Z2,3−4.. Then, thedirect mixing between up-higgsino and bino is dominant and the relevant mixing matrix

139

D. Exotic decay patterns of states for the light singlet scenario

elements can be approximated as

Z1,3−4 + Z2,3−4 = −(

g1v2µu

)2

. (D.10)

Assuming a similar size for the mixing

Z1,3−4!= Z2,3−4 = −

12

(g1v2µu

)2

. (D.11)

this is inserted into equation (D.8) and using the formula for the decay width (D.1) thepartial width becomes

ΓZ→LSP =mz

256π(g2

1 + g22 )g4

1v4

µ4u

√√√√14

−m2χ0

1

m2Z

(m2

Z + 2m2χ0

1

). (D.12)

For BMP5 and BMP6 the partial widths are then 5 · 10−3 MeV and 1 · 10−2 MeV, respectively,well below the experimental uncertainty of the invisible Z boson width. The LSP of BMP4is too heavy to be a decay product of the Z boson.

140

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PUBLICATIONS

The work presented in this thesis is based on research carried out at the Institut für Kern-und Teilchenphysik, Technische Universität Dresden.

The research was performed in collaboration with Prof. Dr. Jan Kalinowski, WojciechKotlarski and Prof. Dr. Dominik Stöckinger. The following publications appeared in theprocess:

[1] Philip Diessner, Jan Kalinowski, Wojciech Kotlarski, and Dominik Stöckinger. “Higgsboson mass and electroweak observables in the MRSSM”. In: JHEP 1412 (2014),p. 124. DOI: 10.1007/JHEP12(2014)124. arXiv: 1410.4791 [hep-ph].

[2] Philip Diessner, Jan Kalinowski, Wojciech Kotlarski, and Dominik Stöckinger. “Ex-ploring the Higgs sector of the MRSSM with a light scalar”. In: JHEP 03 (2016),p. 007. DOI: 10.1007/JHEP03(2016)007. arXiv: 1511.09334 [hep-ph].

[3] Philip Diessner, Jan Kalinowski, Wojciech Kotlarski, and Dominik Stöckinger. “Two-loopcorrection to the Higgs boson mass in the MRSSM”. In: Adv. High Energy Phys.2015 (2015), p. 760729. DOI: 10.1155/2015/760729. arXiv: 1504.05386 [hep-ph].

ACKNOWLEDGMENTS

I thank Dominik Stöckinger for giving me the opportunity to do my Ph.D. studies in his group.He allowed me to pursue the topic in many interesting directions and always provided anopen door when problems needed to be discussed. His supervision and teaching helpedprofoundly to further my understanding of particle physics.

I thank Wojciech H. Kotlarski for being a worthy companion on this journey to a finishedPh.D. thesis. Sharing the moments of learning and frustration while understanding physicsand hunting bugs did not always make it easier but certainly less dull. Thanks for entertain-ing discussions not only concerning particle physics and R-symmetry.

I am thankful to Jan Kalinowski for a fruitful collaboration. Thank you for welcoming meto the institute in Warsaw as well as passing on experience and knowledge of particlephenomenology.

A big thanks goes to each of Markus Bach, Johannes Gube, Sebastian Liebschner, FelixSocher and Heinrich Wilsenach for reading parts of this thesis and providing valuable feed-back and proof reading.

I thank the Research Training Group 1504 for providing interesting lectures and blockcourses as well as giving me the opportunity to travel to schools and conferences.

I am grateful for my family and friends who provide me with support, especially in the lastmonths, and allow me to take restful breaks from the wonders of physics.

This thesis is typeset with the tudscr LATEX class. Feynman diagrams were created usingFeynMP. All figures were drawn using matplotlib and the Python programming language.

ERKLÄRUNG

Hiermit versichere ich, dass ich die vorliegende Arbeit ohne unzulässige Hilfe Dritter undohne Benutzung anderer als der angegebenen Hilfsmittel angefertigt habe; die aus fremdenQuellen direkt oder indirekt übernommenen Gedanken sind als solche kenntlich gemacht.Die Arbeit wurde bisher weder im Inland noch im Ausland in gleicher oder ähnlicher Formeiner anderen Prüfungsbehörde vorgelegt.Die vorliegende Dissertation wurde am Institut für Kern- und Teilchenphysik der TechnischenUniversität Dresden unter wissenschaftlicher Betreuung von Prof. Dr. Dominik Stöckingerangefertigt.

Es haben keine früheren erfolglosen Promotionsverfahren stattgefunden.

Ich erkenne die Promotionsordnung der Fakultät Mathematik und Naturwissenschaften ander Technischen Universität Dresden vom 23.02.2011 in der Fassung vom 18.06.2014 an.

Dresden, 08.08.2016

Philip Dießner