Mass spectrum prediction in non-minimal supersymmetric models Dissertation zur Erlangung des wissenschaftlichen Grades Doctor rerum naturalium (Dr. rer. nat.) vorgelegt von Dipl.-Phys. Alexander Voigt geboren am 14.08.1984 in Dresden Institut für Kern- und Teilchenphysik der Technischen Universität Dresden 2014 Eingereicht am 16.06.2014 Verteidigt am 18.09.2014 1. Gutachter: Prof. Dr. Dominik Stöckinger 2. Gutachter: Prof. Dr. Benjamin Allanach iii Abstract Supersymmetry is an attractive extension of the Standard Model (SM) of particle physics. The minimal supersymmetric extension (MSSM) provides gauge coupling unification, a dark matter candidate particle and can explain the breaking of the electroweak symmetry dynamically. However, it suffers from the little hierarchy and the µ-problem. Non-minimal supersymmetric extensions of the SM with a larger particle content or a higher symmetry can evade the problems of the MSSM. Such models may be well-motivated by Grand Unified Theories (GUTs) and can provide a rich new phenomenology with an extended Higgs sector, exotic particles, additional interactions and a close connection to String Theory. Interesting examples are the Next-to Minimal Supersymmetric Standard Model (NMSSM), which is motivated by the µ-problem, and the Exceptional Supersymmetric Standard Model (E6SSM), which is inspired by E6 GUTs. For phenomenological investigations of supersymmetric (SUSY) models the pole mass spectrum must be calculated from the fundamental model parameters. This task, however, is non-trivial as the spectrum must be consistent with measured low- energy observables (fine-structure constant, Z boson pole mass, muon decay etc.) as well as electroweak symmetry breaking and potential universality conditions on the soft supersymmetry breaking parameters at the GUT scale. Programs, which calculate the SUSY mass spectrum consistent with constraints of this kind are called spectrum generators. In this thesis four different contributions to the prediction of mass spectra and model parameters in non-minimal SUSY models are presented. (i) One-loop matching corrections of the E6SSM gauge and Yukawa couplings to the SM are calculated to increase the precision of the mass spectrum prediction in the constrained E6SSM. (ii) The β-functions of vacuum expectation values (VEVs) are calculated in a general and supersymmetric gauge theory at the one- and two-loop level. The results enable an accurate calculation of the renormalization group running of the VEVs in non-minimal SUSY models. (iii) An NMSSM extension of Softsusy, a spectrum generator for the MSSM, is implemented. It represents a precise alternative to the already existing spectrum generator NMSPEC. (iv) FlexibleSUSY is presented, a general framework which creates a fast, modular and precise spectrum generator for any user-defined SUSY model. It represents a generalization of the hand-written SUSY spectrum generators and allows the study of a large variety of new SUSY models easily with high precision. v Contents 1 Introduction 1 2 Supersymmetry 5 2.1 The minimal supersymmetric standard model (MSSM) . 6 2.1.1 Advantages of the MSSM . 7 2.1.2 Problems of the MSSM . 7 2.2 Non-minimal supersymmetric models . 8 2.2.1 Examples for non-minimal models . 9 2.2.2 Advantages of non-minimal models . 9 2.2.3 Problems of non-minimal models . 10 3 Calculation of low-energy gauge and Yukawa couplings in the E6SSM 13 3.1 Motivation . 13 3.2 Calculation of the Yukawa couplings . 16 3.3 Implementation into a CE6SSM spectrum generator . 17 3.4 Numerical evaluation and application . 20 3.4.1 Reduced matching scale dependency . 20 3.4.2 Parameter scan . 21 3.4.3 Parameter space exclusion . 23 3.4.4 Updating benchmark points from the literature . 25 3.5 Summary and conclusion . 26 4 Renormalization of vacuum expectation values 29 4.1 Motivation . 29 4.2 Calculation of the VEV counter-term . 30 4.2.1 Background field method . 30 4.2.2 Lagrangian . 32 4.2.3 Renormalization . 34 4.2.4 Calculation of the anomalous dimensions . 38 4.3 Result for the VEV β-function . 39 4.3.1 General gauge theory . 39 4.3.2 Kinetic mixing . 40 4.3.3 Supersymmetric gauge theory . 41 4.4 Application to the MSSM and NMSSM . 43 4.4.1 MSSM . 43 4.4.2 NMSSM . 44 4.5 Summary and conclusion . 46 vii 5 Next-to-minimal Softsusy 47 5.1 Motivation for a NMSSM extension of Softsusy . 47 5.2 The NMSSM . 48 5.3 Physical problem statement . 49 5.4 Implementation . 51 5.4.1 Class hierarchy . 51 5.4.2 Implementation of the β-functions . 55 5.4.3 Solving the boundary value problem . 57 5.4.4 Calculation of the pole mass spectrum . 60 5.5 Tests and comparison to NMSPEC . 61 5.6 Summary and conclusion . 63 6 FlexibleSUSY – A spectrum generator generator 65 6.1 Motivation . 65 6.2 General physical problems . 67 6.2.1 Definition of the model . 68 6.2.2 Boundary conditions . 69 6.2.3 Calculation of pole masses . 71 6.3 Implementation . 71 6.3.1 Creating the C++ source code . 72 6.3.2 Class hierarchy . 77 6.3.3 Boundary conditions and the FlexibleSUSY model file . 82 6.3.4 Solving the boundary value problem . 91 6.3.5 Calculation of the pole masses . 95 6.3.6 Summary . 96 6.4 Applications . 98 6.4.1 Run-time comparison . 98 6.4.2 Comparison of Higgs masses in supersymmetric models . 100 6.4.3 Influence of loop corrections of different order on the pole mass spectrum . 109 6.4.4 Influence of GUT threshold corrections on the Higgs mass . 112 6.5 Summary and conclusion . 115 7 Summary 117 A Group invariants 121 B MSSM field content 123 C The E6SSM 125 C.1 Gauge group . 125 C.2 Field content . 125 C.3 Superpotential . 127 C.4 The constrained E6SSM . 129 viii C.5 Mass eigenstates . 130 C.5.1 Higgs sector . 130 C.5.2 Inert Higgs sector . 131 C.5.3 Survival Higgs sector . 132 C.6 Self-energies . 132 C.6.1 W boson . 132 C.6.2 Fermions . 134 C.7 Gauge coupling threshold corrections . 136 D NMSSM SLHA input template file 139 E CMSSM speed test SLHA input template file 141 Bibliography 143 ix Chapter 1 Introduction The physics of elementary particles aims for a unified description of the fundamental constituents and forces in the universe. The Standard Model (SM) of particle physics, which was supported by the discovery of a Higgs boson in 2012, is a very successful description of all known particles as well as the electromagnetic, weak and strong interactions. However, the SM does neither describe gravity nor does it contain particles which are candidates for non-baryonic dark matter. In addition, it suffers from the hierarchy problem and fails to correctly explain observations such as the anomalous magnetic moment of the muon for example. Supersymmetric (SUSY) extensions of the SM can provide solutions to all of these problems. Thereby the space-time symmetries are non-trivially extended so that fundamental relations be- tween fermions and bosons arise. A supersymmetric theory must therefore contain as many fermionic degrees of freedom as bosonic ones. For this reason supersymmetric extensions of the SM introduce many new bosonic and fermionic fields, which leads to a rich and interesting phenomenology. The Minimal Supersymmetric Standard Model (MSSM) has been extensively stud- ied in the past decades. It provides gauge coupling unification, which allows a unifi- cation of the three forces, contains a non-baryonic dark matter candidate particle and can explain the origin of the electroweak symmetry breaking. However, the MSSM is not without problems as well: Parameter scenarios which lead to the measured value of the Higgs mass often re-introduce the hierarchy problem. Furthermore, it contains the dimensionful parameter µ, whose value is well-motivated to be of the order of the gauge coupling unification scale, O(1016) GeV, as well as of the order of the electroweak scale, O(102) GeV. This tension is called the µ-problem. Non-minimal supersymmetric models with a larger particle content or a higher sym- metry can solve the problems of the MSSM. Such models may be well-motivated by Grand Unified Theories (GUTs) and can provide a rich new phenomenology with an extended Higgs sector, exotic particles, additional interactions and close connections to String Theory. Interesting examples are the Next-to Minimal Supersymmetric Standard Model (NMSSM) and the Exceptional Supersymmetric Standard Model (E6SSM). The former extends the Higgs sector of the MSSM by an additional singlet field to solve the µ-problem. The E6SSM is motivated by E6 GUTs and provides a matter particle content which fits into fundamental representations of the E6. In addition it introduces an extra U(1)N gauge symmetry, which leads to a new gauge boson Z′. 1 Chapter 1 Introduction For phenomenological investigations of supersymmetric models many complicated computations must be performed. Given the Lagrangian of the model, the first step is usually to calculate the pole mass spectrum given a set of fundamental theory input parameters. This mass spectrum can then used to calculate further observables, such as for example particle decays, cross sections or the dark matter relic density. These observables can be compared with measurements in phenomenological studies to search for allowed and excluded model parameter space regions. The pole masses and the decays may as well be used in event generators and detector simulations to study how decays of SUSY particles manifest in a detector. The calculation of the pole mass spectrum in a SUSY model is a non-trivial task: The model is required to correctly predict the measured low-energy observables, such as the fine-structure constant, the Z boson pole mass and the muon decay for example.