Faculty of Science Physics and Astronomy January 12, 2017 Experimental and Theoretical Constraints on pMSSM Models Investigating the diboson excess and fine-tuning Ruud Peeters Supervisor: Dr. Sascha Caron Acknowledgements This thesis would not have been possible without the help of many people. First and foremost, I want to thank my supervisor Sascha Caron, whose unlimited enthusiasm I will never forget. I also want to thank my unofficial second supervisor Wim Beenakker, who always had an answer to my questions. I want to thank Ronald Kleiss for agreeing to be the second corrector of this thesis. I want to thank Krzyszof Rolbiecki and Jong Soo Kim for their help with the detector simulation in the diboson analysis and Roberto Ruiz de Austri for his implementation of the fine-tuning measure in SoftSUSY and the many discussions about the correct implementation. A big thanks goes out to Melissa van Beekveld, her help with the physics, programming and dealing with supervisors was invaluable. I would like to thank Melissa, Bob and Milo for reading parts of my thesis and making sure that some horrible mistakes, typos and other errors did not make the final version. Finally I want to thank my family for their support during this research. 1 Contents 1 Introduction 4 2 The Standard Model6 2.1 Relativistic Lagrangian mechanics..........................6 2.2 Symmetries.......................................7 2.3 The Higgs mechanism.................................8 2.4 Symmetries of the Standard Model..........................9 2.5 Fermionic particle content............................... 11 2.6 Problems of the Standard Model........................... 12 2.6.1 Dark matter.................................. 12 2.6.2 The hierarchy problem............................. 13 3 Supersymmetry 15 3.1 Idea behind supersymmetry.............................. 15 3.2 Supersymmetry breaking................................ 16 3.3 The MSSM....................................... 16 3.3.1 Mixing in the MSSM.............................. 17 3.4 The pMSSM...................................... 18 3.5 Conclusion....................................... 20 4 The diboson excess 21 4.1 Collider experiments.................................. 21 4.1.1 Variables used in collider physics....................... 22 4.2 The diboson excess................................... 23 4.2.1 Event selection................................. 23 4.3 A diboson excess with pMSSM models........................ 26 4.3.1 The Galactic Centre excess models...................... 26 4.3.2 pMSSM processes with diboson creation................... 27 4.4 The optimal GCE model................................ 30 4.5 Simulation........................................ 32 4.6 Analysis......................................... 32 4.7 pMSSM event selection................................ 33 4.8 Results.......................................... 34 4.8.1 The best parameter values........................... 35 4.8.2 Detector simulation.............................. 38 4.9 Conclusion....................................... 40 5 Fine-tuning in pMSSM models 42 5.1 Theoretical background................................ 43 5.1.1 Renormalisation................................ 43 5.1.2 SUSY Higgs mechanism............................ 43 2 5.2 Quantifying fine-tuning................................ 46 5.2.1 Measures of fine-tuning............................ 47 5.3 Fine-tuning in the literature.............................. 50 5.3.1 Requirements for minimal fine-tuning.................... 51 5.3.2 Natural SUSY................................. 52 5.4 Calculating fine-tuning................................. 52 5.5 Fine-tuning scan.................................... 53 5.6 Results.......................................... 54 5.6.1 Final results.................................. 56 5.7 Discussion........................................ 58 5.8 Conclusion....................................... 59 5.9 Outlook......................................... 60 A Minimisation of the SUSY Higgs potential 61 Bibliography 63 3 Chapter 1 Introduction Research in elementary particle physics has been going on for a long time. The current status is that there is a theory, the Standard Model of particle physics, that can explain almost all processes we observe. However, there are some experimental observations and theoretical problems that show that the Standard Model is not complete. There are many different theories that try to address these problems, but none has been experimentally verified. In this thesis, one of these beyond the Standard Model theories will be examined in more detail. This is the theory of supersymmetry, which introduces one (or more) new particles for each particle in the Standard Model. This thesis consists of two different analyses. The goal of both analyses is to investigate the viability of supersymmetry, with a focus on the dark matter properties of supersymmetric models. The analyses have a completely different approach however. In the first analysis, an excess in the ATLAS detector at CERN is investigated. This excess might be a signal of the production of supersymmetric particles. A specific set of supersymmetric models is used to find out if this excess can be caused by supersymmetric processes. The other research project is to study fine-tuning in supersymmetry. Fine-tuning is a measure of how (un)natural a theory is. An unnatural theory is a theory that only works if the parameters of the theory are very restricted, without a clear explanation. Such a theory is not very credible. The goal of this analysis is to find the most natural supersymmetric models, that also satisfy all experimental constraints. The structure of this thesis is as follows. Chapter2 gives a theoretical background of the Standard Model of particle physics. Supersymmetry is introduced as a beyond the Standard Model theory in Chapter3. The analysis of the diboson excess in the ATLAS detector is discussed in Chapter4. The last chapter, Chapter5, treats fine-tuning in supersymmetry. 4 Conventions • Natural units are used throughout this thesis, so ~ = c = 1. Masses will therefore be given in units of energy, generally in GeV or TeV. • The Einstein sum convention is used in this thesis, meaning that all repeated indices are summed over. @ • A partial derivative @xµ is written as @µ. • The Dirac gamma matrices γµ are defined as: ! i! 0 0 I2 i 0 σ γ = ; γ = i ; I2 0 −σ 0 i where I2 denotes the 2x2 identity matrix and σ are the Pauli matrices, defined by: ! ! ! 0 1 0 −i 1 0 σ1 = ; σ2 = ; σ3 = 1 0 i 0 0 −1 • The slashed notation a= denotes the contraction of a four-vector aµ with the gamma matrices γµ: µ a= = γ aµ • The adjoint of a fermionic field is defined as: = yγ0 5 Chapter 2 The Standard Model In this chapter, a quick overview of the Standard Model of particle physics is given. The Lagrangian is introduced first, then the symmetries of the Standard Model Lagrangian are discussed. The Higgs mechanism is introduced, followed by a description of the bosonic and fermionic particle content of the Standard Model. Finally, some problems of the Standard Model are discussed. This section is mainly based on [1] and [2]. An overview of group theory can be found in [3]. All other sources are cited in the text. 2.1 Relativistic Lagrangian mechanics Quantum field theory describes the world of elementary particles, combining quantum mechanics and special relativity in one theory. The most important quantity in quantum field theory is the Lagrangian (L).∗ The Lagrangian is so important because the equations of motion of all particles can be deduced from it. There are three different kinds of particles in the Standard Model, each with their own terms in the Lagrangian: the scalar particles (spin-0), fermions (spin-1/2) and vector bosons (spin-1). In this chapter, a general scalar, fermion and vector boson will be represented by φ, and Aµ respectively. An implicit spacetime dependence is assumed for all fields (φ = φ(xµ)). The terms in the Lagrangian can be subdivided into three groups: kinetic terms, mass terms and interaction terms. The kinetic terms of the three different kinds of particles are shown in Equation 2.1. µ ∗ Scalar: (@µφ)(@ φ ) Fermion: i @=µ (2.1) µν Vector boson: FµνF µ In this equation, γ are the four-dimensional Dirac matrices and Fµν = @µAν − @νAµ. The kinetic terms dictate the behaviour of a massless particle without interactions. ∗This is actually the Lagrangian density, but it is usually called the Lagrangian. The same will be done in this thesis. 6 However, most particles in the Standard Model are not massless. They have mass terms of the following form: Scalar: m2φφ∗ Fermion: i m (2.2) 2 µ Vector boson: m AµA Interactions make up the third set of terms, dictating which interactions are possible in a theory. An example of an interaction term is L = λ φψ, which indicates an interaction between a scalar and two fermions. The interaction strength is given by λ. The equations of motion can be derived from the Lagrangian using the Euler-Lagrange equation for all fields in the theory: δL δL = @µ : (2.3) δφ δ(@µφ) 2.2 Symmetries A Lagrangian can have one or more symmetries. These are operations that can be applied to the fields in the theory while leaving the Lagrangian invariant. This section will focus on continuous symmetries. The most elementary example of such a symmetry is a global phase transformation: a phase transformation that does not depend on the spacetime coordinate. A global phase transformation is an element of the U(1) group and it transforms