Renormalization Group Invariants in the Minimal Supersymmetric Standard Model Master’s Thesis September 2014 Tom van Daal [email protected] Supervisor: Prof. dr. W.J.P. Beenakker Department of Theoretical High Energy Physics Institute for Mathematics, Astrophysics and Particle Physics Radboud University Nijmegen Contents 1 Introduction 5 1.1 Conventions........................................ 5 2 The Standard Model 7 2.1 IntroductiontotheStandardModel . 7 2.2 TheStandardModelfieldcontent. 7 2.3 The Standard Model Lagrangian . 8 2.4 Flavormixing....................................... 10 3 Symmetries of the Standard Model 15 3.1 Noether’stheorem .................................... 15 3.2 Poincar´esymmetries ................................... 16 3.3 Gaugesymmetries .................................... 16 3.4 Flavorsymmetries .................................... 17 3.4.1 Flavor group without flavor mixing . 17 3.4.2 Flavor group with flavor mixing . 21 4 Supersymmetry 25 4.1 Introductiontosupersymmetry . 25 4.2 Motivation for supersymmetry . 26 4.3 Supersymmetric Lagrangians . 29 4.3.1 Chiralsupermultiplets . 29 4.3.2 Gaugesupermultiplets.............................. 30 4.3.3 Chiral and gauge supermultiplets combined . 31 4.4 Supersymmetrybreaking................................. 32 5 The Minimal Supersymmetric Standard Model 33 5.1 TheMSSMfieldcontent................................. 33 5.2 R-parity.......................................... 34 5.3 The MSSM Lagrangian . 34 5.4 Softsupersymmetrybreaking . 37 5.5 Countingindependentparameters . 38 5.6 SymmetriesoftheMSSM ................................ 42 6 Renormalization group techniques 45 6.1 What is renormalization? . 45 6.2 Renormalization group equations . 46 6.3 E↵ectivefieldtheories .................................. 46 6.4 Renormalization group invariants . 47 7 One-loop β-functions for MSSM scalar masses 51 7.1 The propagator for scalar fields . 51 7.1.1 Higgs doublets . 52 7.1.2 Scalarfields.................................... 52 7.2 One-loop corrections to scalar propagators . 53 7.2.1 Type 1 diagrams . 54 7.2.2 Type 2 diagrams . 55 7.2.3 Type 3 diagrams . 56 7.2.4 Type 4 diagrams . 58 7.3 Renormalization of scalar fields . 59 7.3.1 Field renormalization . 59 7.3.2 Mass renormalization . 60 7.4 Results........................................... 61 3 8 RG invariants in the MSSM 62 8.1 Constructing RG invariants in the canonical way . 62 8.2 RG invariants from a di↵erentperspective . 65 9 Conclusion 68 AIndices 69 B Unitary matrices 70 C Spinors 72 C.1 Diracspinors ....................................... 72 C.2 Weylspinors ....................................... 72 D Superpotential contributions to the MSSM Lagrangian 74 E Integrals 78 F One-loop mass counterterms for MSSM scalar fields 83 List of abbreviations 87 Acknowledgements 88 Bibliography 90 4 1 Introduction The experiments at the Large Hadron Collider (LHC) at CERN, among other experiments, are searching for physics beyond the Standard Model of particle physics, since we know that the Standard Model cannot be the final story. Experimentalists are looking for signals that do not fit well in the Standard Model framework. One job for theorists, on the other hand, is to come up with new models that describe physics at even smaller length scales (or, equivalently, higher energy scales) that solve the current problems with the Standard Model. Over the last decades, many new theories have been described in literature, all coming with both pros and cons. In this thesis we will focus on one particular beyond the Standard Model theory: the so-called Minimal Supersymmetric Standard Model (MSSM). This is the most minimal extension to the Standard Model that respects an additional spacetime symmetry, called supersymmetry. In short, the MSSM predicts that each Standard Model particle has a supersymmetric counterpart with exactly the same quantum numbers and mass (except for the spin that di↵ers by half a unit). We already know that even the MSSM could not be the “theory of everything” though. Namely, since supersymmetric particles have not been found yet, this means that either supersymmety is simply not implemented in nature, or that supersymmetric particles are much heavier (at least beyond the reach of the LHC) than their Standard Model counterparts, i.e. supersymmetry is a broken symmetry of nature. In this thesis, of course, we will assume the latter. At energy scales many orders of magnitude higher than the typical LHC collider scale of (10 TeV), even beyond the range of validity of the MSSM, new physics would provide boundary conditionsO to the MSSM. Say that supersymmetric particles are discovered at the LHC and that some relevant new parameters are measured, would there then be a way to check which of those boundary conditions could be realized in nature? The answer is yes: quite recently, a new method has been devised that could probe these boundary conditions by making clever use of so-called renormalization group (RG) invariants using low energy data only (i.e. the new data coming from particle accelerators). Constructing these RG invariants as probes for high scale physics models can be a very tedious and complicated job though, and this has not been done yet for (the scalar sector of) the MSSM, but only for heavily constrained supersymmetric models (in these models typically about 80% of the MSSM parameters are “thrown away”). In this thesis we will attempt to construct RG invariant quantities for the full MSSM, i.e. without any simplifications. In this thesis a thorough analysis of symmetries of the MSSM will be performed. Most of this analysis will be carried out for the Standard Model though (as this theory is a bit simpler) and the results will then be generalized to the MSSM. Finally, an attempt will be made to unify symmetries of the MSSM with the concept of RG invariant quantities. If such a relation would turn out to exist, then it would be much easier to tell for a given (supersymmetric) theory if RG invariants could exist at all, and, if the answer is positive, how to construct them. After all, simplifying a certain theory (i.e. setting parameters equal to each other or to zero) generally means that symmetry is added to its Lagrangian. Could this in turn mean that also the number of RG invariants increases? In the first section we will discuss the Standard Model, mostly focussing on its Lagrangian and mixing of fermionic flavors. In section 2 we will present a thorough analysis of symmetries of the Standard Model. The obtained results for family space will be most important for the remainder of this thesis. Subsequently, the basic concepts of supersymmetry will be discussed and a general supersymmetric Lagrangian will be constructed. In the next section we will consider the MSSM and perform a careful count of its independent parameters. Also the symmetry analysis for the Standard Model will then be generalized to the MSSM. In the subsequent section, the concept of renormalization and the utility of RG invariants will be discussed. Section 7 will be the most technical part of this thesis. Here the one-loop RG equations for the MSSM scalar masses will be derived. Finally, an attempt will be made to construct RG invariants in the MSSM from these obtained RG equations and we will consider the role of symmetries in this context. 1.1 Conventions Throughout this thesis we will make use of the following conventions: We adopt natural units where ~ = c = 1. With this choice, mass is the only dimension that • 5 is left, i.e.: 1 1 [mass] = [energy] = [length]− =[time]− , (1) where the square brackets denote the mass dimension. We make use of the Einstein summation convention, which means that all repeated indices • are summed over (unless explicitly stated otherwise). The various types of indices that are used in this thesis are defined in appendix A. For the Minkowski metric ⌘ we choose the convention ⌘ = diag(1, 1, 1, 1). • µ⌫ µ⌫ − − − We define the contravariant spacetime and energy-momentum four-vectors to be • xµ =(t, ~x) ,pµ =(E,p~ ) , (2) and the (ordinary) derivative is defined as @ @ . (3) µ ⌘ @xµ The n n identity matrix will be denoted by In, and the most important matrices in family • space will⇥ be indicated in boldface (to make the text more readable). It is important to be aware of the fact that the sections on the Standard Model make use • of Dirac spinors to describe the fermionic fields, while the sections on supersymmetry and renormalization group techniques use Weyl spinors instead. The di↵erences between Dirac and Weyl spinors are explained in appendix C. The abbreviation “h.c.” is short for “Hermitian conjugate”. • 6 2 The Standard Model The Standard Model of particle physics incorporates all known fundamental particles and their (non-gravitational) interactions and forms the heart of particle physics. In this section we briefly review the main aspects of the Standard Model. We will discuss its particle content and provide the most important mathematical structures. Much more on the Standard Model can be found in any textbook on quantum field theory or particle physics, e.g. in [1]. 2.1 Introduction to the Standard Model The Standard Model is a quantum field theory that describes the dynamics of all known funda- mental particles up to an incredible accuracy. It provides the framework for three out of the four (known) fundamental forces of nature: the electromagnetic, weak and strong interactions. The Standard Model was completed in the early 1970s, but not fully experimentally verified until the discovery of the Higgs boson at the Large Hadron Collider (LHC) in 2012 [2, 3]. Despite the great success of the Standard Model, there are also a couple of (significant) short- comings. One of them being the complete absence of gravity. It is expected that at the Planck scale gravity becomes comparable in strength to the other forces, which implies that quantum gravity e↵ects can no longer be ignored1. In other words, the Standard Model breaks down and a reconciliation of general relativity and quantum mechanics is needed. This is why the Standard Model cannot be the theory of everything; it provides an excellent e↵ective framework for “low” energy physics, but for higher energies (i.e.