Supersymmetric Grand Unification Sander Mooij Master’s Thesis in Theoretical Physics Institute for Theoretical Physics University of Amsterdam Supervisor: Prof. Dr. Jan Smit August 18, 2008 Contents 1 The Standard Model 6 1.1 Introduction to the Standard Model . 6 1.1.1 Fundaments . 6 1.1.2 Contents of the Standard Model . 10 1.1.3 Writing down the SM Lagrangian . 12 1.2 Troubles and shortcomings of the Standard Model . 18 1.2.1 Reasons to go beyond the SM . 18 1.2.2 Righthanded neutrinos . 18 1.2.3 Renormalization and the hierarchy problem in the SM . 20 1.3 Running of Coupling Constants: a first clue for Grand Unification . 21 1.3.1 Renormalized Perturbation Theory . 22 1.3.2 Calculation of counterterms . 23 1.3.3 Callan-Symanzik equation and renormalization equation . 25 1.3.4 Running Coupling Constants . 26 2 Group Theoretical Backgrounds 31 2.1 Roots . 31 2.1.1 Simple roots . 32 2.1.2 Dynkin labels . 33 2.2 Weights and representations . 34 2.3 Applications . 35 2.3.1 Casimir operators . 35 2.3.2 Eigenstates . 36 2.3.3 Decomposition of tensor products . 36 2.3.4 Branching rules and projection matrices . 38 3 Grand Unification 40 3.1 Embedding the SM fields in SU(5) . 40 3.2 Writing down the SU(5) Lagrangian . 45 3.2.1 Fermion kinetic terms . 45 3.2.2 Fermion mass terms . 46 3.2.3 Gauge kinetic terms . 48 3.3 The Higgs mechanism in SU(5) . 48 3.4 Consequences of SU(5) unification . 50 3.5 SO(10) unification: embedding of SM fields . 51 3.5.1 Exploring the 16 rep . 51 3.5.2 Branching to the SM . 52 1 3.5.3 Ordering of generators and fields . 55 3.6 The SO(10) Higgs mechanism . 57 3.7 SO(10): a short summary . 58 4 Supersymmetry and the Minimal Supersymmetric Standard Model 60 4.1 Introduction to supersymmetry . 60 4.1.1 A new symmetry . 60 4.1.2 Superfields . 62 4.2 Construction of the MSSM . 63 4.2.1 The rules of the game . 63 4.2.2 Matter kinetic terms . 65 4.2.3 Higgs kinetic terms . 67 4.2.4 Gauge kinetic terms . 67 4.2.5 Superpotential terms . 68 4.3 How sparticles solve the hierarchy problem . 70 4.4 Breaking of Supersymmetry . 71 4.5 Running of the Coupling Constants in the MSSM: a true clue for Grand Unification . 73 5 SUSY SO(10) 76 5.1 The situation at the GUT-scale . 76 5.2 Down from the GUT scale to the electroweak scale . 77 6 Yukawa unification in SUSY SO(10) 79 6.1 Experimental input at the electroweak scale . 79 6.1.1 No mixing . 81 6.1.2 Mixing . 81 6.2 Standard Model Yukawa running . 82 6.2.1 SM Renormalization Group Equations . 82 6.2.2 Results . 83 6.2.3 Conclusions . 84 6.3 MSSM Yukawa running . 85 6.3.1 MSSM Renormalization Group Equations . 86 6.3.2 Results . 86 6.3.3 Conclusions . 91 7 Dermisek and Raby’s family symmetric approach 92 7.1 Construction of DR GUT-scale Yukawa matrices . 92 7.1.1 D3 ................................. 92 7.1.2 The DR superpotential . 94 7.1.3 DR Dirac Yukawa matrices . 95 7.1.4 χ2 analysis . 95 7.2 Analysis of DR Yukawa matrices . 96 8 Conclusions 98 References . 100 2 A Conventions 102 A.1 Essentials . 102 A.2 SU(2) and SU(3) . 104 B Associating LH fields to RH fields 105 C Additional SM symmetries 107 3 Abstract At an energy scale around 1016 GeV the Standard Model (SM) is assumed to give way to a theory in which all scalar, spinor and boson fields are in one universal gauge group, a so-called Grand Unified Theory (GUT). In this thesis we review the construction of supersymmetric GUTs and check in Mathematica how well its predictions, gauge coupling unification and Yukawa coupling unification, are met. 4 Thesis Outline To provide a solid background we first review the Standard Model and its prob- lematic features, especially the hierarchy problem. In the second chapter some necessary group theory, mainly weight vector analysis, is studied. Then, in the third chapter, we construct the two most popular GUTs, SU(5) and SO(10). We show that in these theories the SM hierarchy problem is still unresolved and see that gauge couplings do not meet well enough to fit low-energy measurements. There- fore, we are led to study supersymmetric GUTs. In the fourth chapter we introduce supersymmetry (SUSY). We review the minimal supersymmetric extension of the Standard Model (MSSM) and see how it solves the aforementioned problems. The theoretical part of the thesis is concluded by the construction of the most simple (elegant) SUSY GUT. In the phenomenological part we convert electroweak scale observables, particle masses and quark and lepton mixing matrices to Yukawa matrices. Two-loop Yukawa Renormalisation Group Equations are solved in Mathematica. This en- ables us to check whether the SUSY GUT promise of Yukawa unification is met. In the last chapter a recent model by Dermisek and Raby that connects to the results of this Yukawa analysis is studied. 5 Chapter 1 The Standard Model Dramatic progress in (special) relativity, quantum mechanics, quantum field the- ory and quantumchromodynamics has led us to the Standard Model of Elementary Particle Physics, or SM. It emerged in the early seventies and has been proven right in numerous experiments afterwards. The SM lists all elementary particles and describes their interactions via the strong, weak interaction and U(1) “hyper” interaction. After spontaneous symmetry break- ing, caused by the so-called Higgs boson taking a nonzero vacuum expectation value, the weak and U(1) hyperforce combine into the electromagnetic interaction. In a minute we will see how exactly this comes to be. 1.1 Introduction to the Standard Model In this first section we will construct the full Standard Model Lagrangian. 1.1.1 Fundaments The SM Lagrangian rests on the fundamental assumptions that it should be gauge invariant, Lorentz invariant and renormalizable. SU(3) × SU(2) × U(1) gauge invariance To every field f appearing in the SM we associate a gauge transformation: a a b b f → eiα Ts eiβ Tw eiγY f. (1.1) a a The first part eiα Ts is the SU(3) part. By definition, SU(3) has 8 generators. a These are the 8 matrices Ts . We can choose (infinitely) many representations for a b abc c these generators, as long as they obey the commutation relations [Ts ,Ts ] = if Ts . (All over this work, a sum over repeated indices is assumed.) The structure con- stants are typical for SU(3), they define the group. They can be looked up in appendix A.2. The 8 αa are infinitesimal displacement vectors specifying the gauge transforma- tion. Interesting possibilities are the so-called fundamental representation, where we take a a (1/2 times) the 8 traceless 3 × 3 Gell-Mann matrices λ to represent the Ts , the 6 a trivial representation where we set all Ts to 0 so that this whole part disappears from the gauge transformation, and the adjoint representation where we have 8 a 8 × 8 matrices (Ti)jk = −ifijk that represent the Ts . The connection with physics now is that the strong force is mediated by the 8 gauge a bosons (gluons) Gµ that transform in the adjoint representation of SU(3). Every fermion that is sensitive to the strong force (that is, every quark) is represented by a field that transforms in the fundamental representation of SU(3), while to the antiquarks we associate a field that transforms in the antifundamental representa- a λa ? λa 1 tion: Ts = −( 2 ) ≡ 2 . Fields representing particles that do not feel the strong interaction (particles that cannot couple to a gluon, leptons, that is) transform in the trivial representation of SU(3). b b The second part eiβ Tw is the SU(2) part. Everything works out in essentially the same way as in the SU(3) case. The main difference is that SU(2) has only 3 gen- b b erators Tw, so there will be only 3 gauge bosons Aµ. These three bosons mediate b σb b the weak force. The fundamental representation is given by Tw = 2 , where the σ are the well-known Pauli matrices. As we have σ2σbσ2 = σb, the antifundamental representation of SU(2) is equivalent to its fundamental representation. Such a representation is called real. The third part, eiγY , that contains the U(1) part is different in the sense that there is just one (one-dimensional) generator Y involved. Y is just a number and is called hypercharge. Such a gauge theory with just one generator (and, therefore, just one gauge boson Bµ) is called Abelian while a gauge theory with several non- commuting generators is non-Abelian. Reps of non-Abelian groups are labeled by their dimension: the fundamental rep of SU(3) for example, where generators are 3 × 3 matrices) for example is denoted as 3, the antifundamental rep as 3 and the trivial rep as 1. The reps of the Abelian group U(1) are simply denoted by the value of the one dimensional generator Y , the hypercharge. Now for gauge invariance. Of each particle we know what forces it is subject to. So, for each of the associated fields we know the representation, the form of a b the matrices Ts , Tw, Y , in which it transforms.