Supersymmetry Demystified: A Self-Teaching Guide (Demystified Series)

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Supersymmetry Demystified
Patrick Labelle, Ph.D.
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Acknowledgments xiii
CHAPTER 1 Introduction 1 1.1 What Is Supersymmetry, and Why Is It
Exciting (the Short Version)? 1 1.2 What This Book Is and What It Is Not 3 1.3 Effective Field Theories, Naturalness,
and the Higgs Mass 6 1.4 Further Reading 11
CHAPTER 2 A Crash Course on Weyl Spinors 13 2.1 Brief Review of the Dirac Equation
and of Some Matrix Properties 14 2.2 Weyl versus Dirac Spinors 17 2.3 Helicity 22 2.4 Lorentz Transformations and Invariants 24 2.5 A First Notational Hurdle 26 2.6 Building More Lorentz Invariants
Out of Weyl Spinors 28 2.7 Invariants Containing Lorentz Indices 32 2.8 A Useful Identity 33 2.9 Introducing a New Notation 34 2.10 Quiz 40
vi Supersymmetry Demystified
CHAPTER 3 New Notation for the Components of Weyl Spinors 43 3.1 Building Lorentz Invariants 51 3.2 Index-Free Notation 53 3.3 Invariants Built Out of Two Left-Chiral Spinors 56 3.4 The ε Notation for ±iσ 2 57 3.5 Notation for the Indices of σμ and σ μ 60 3.6 Quiz 65
CHAPTER 4 The Physics of Weyl, Majorana, and Dirac Spinors 67 4.1 Charge Conjugation and Antiparticles
for Dirac Spinors 68 4.2 CPT Invariance 70 4.3 A Massless Weyl Spinor 72 4.4 Adding a Mass: General Considerations 73 4.5 Adding a Mass: Dirac Spinors 74 4.6 The QED Lagrangian in Terms of Weyl Spinors 76 4.7 Adding a Mass: Majorana Spinors 78 4.8 Dirac Spinors and Parity 81 4.9 Definitionof the Masses of Scalars
and Spinor Fields 82 4.10 Adding a Mass: Weyl Spinor 85 4.11 Relation Between Weyl Spinors
and Majorana Spinors 86 4.12 Quiz 88
CHAPTER 5 Building the Simplest Supersymmetric Lagrangian 89 5.1 Dimensional Analysis 90 5.2 The Transformation of the Fields 91 5.3 Transformation of the Lagrangian 95 5.4 Quiz 99
Contents vii
CHAPTER 6 The Supersymmetric Charges and Their Algebra 101 6.1 Charges: General Discussion 102 6.2 Explicit Representations of the Charges
and the Charge Algebra 107 6.3 Finding the Algebra Without the
Explicit Charges 111 6.4 Example 113 6.5 The SUSY Algebra 115 6.6 Nonclosure of the Algebra for
the Spinor Field 125 6.7 Introduction of an Auxiliary Field 128 6.8 Closure of the Algebra 131 6.9 Quiz 131
CHAPTER 7 Applications of the SUSY Algebra 133 7.1 Classification of States Using the Algebra:
Review of the Poincare Group 134 7.2 Effects of the Supercharges on States 140 7.3 The Massless SUSY Multiplets 144 7.4 Massless SUSY Multiplets and the MSSM 146 7.5 Two More Important Results 147 7.6 The Algebra in Majorana Form 150 7.7 Obtaining the Charges from Symmetry
Currents 151 7.8 Explicit Supercharges as Quantum
Field Operators 153 7.9 The Coleman-Mandula No-Go Theorem 156 7.10 The Haag-Lopuszanski-Sohnius Theorem
and Extended SUSY 157 7.11 Quiz 158
CHAPTER 8 Adding Interactions: The Wess-Zumino Model 159 8.1 A Supersymmetric Lagrangian With Masses
and Interactions 160 8.2 A More General Lagrangian 167
viii Supersymmetry Demystified
8.3 The Full Wess-Zumino Lagrangian 173 8.4 The Wess-Zumino Lagrangian in
Majorana Form 175 8.5 Quiz 176
CHAPTER 9 Some Explicit Calculations 177 9.1 Refresher About Calculations of Processes
in Quantum Field Theory 178 9.2 Propagators 182 9.3 One Point Function 185 9.4 Propagator of the B Field to One Loop 189 9.5 Putting It All Together 201 9.6 A Note on Nonrenormalization Theorems 203 9.7 Quiz 204
CHAPTER 10 Supersymmetric Gauge Theories 205 10.1 Free Supersymmetric Abelian Gauge Theory 206 10.2 Introduction of the Auxiliary Field 209 10.3 Review of Nonabelian Gauge Theories 211 10.4 The QCD Lagrangian in Terms
of Weyl Spinors 217 10.5 Free Supersymmetric Nonabelian
Gauge Theories 219 10.6 Combining an Abelian Vector Multiplet
With a Chiral Multiplet 222 10.7 Eliminating the Auxiliary Fields 232 10.8 Combining a Nonabelian Gauge Multiplet
With a Chiral Multiplet 233 10.9 Quiz 237
CHAPTER 11 Superspace Formalism 239 11.1 The Superspace Coordinates 240 11.2 Example of Spacetime Translations 242
Contents ix
11.3 Supersymmetric Transformations of the Superspace Coordinates 244
11.4 Introduction to Superfields 248 11.5 Aside on Grassmann Calculus 250 11.6 The SUSY Charges as Differential Operators 255 11.7 Constraints and Superfields 261 11.8 Quiz 268
CHAPTER 12 Left-Chiral Superfields 269 12.1 General Expansion of Left-Chiral Superfields 269 12.2 SUSY Transformations of the
Component Fields 273 12.3 Constructing SUSY Invariants Out of
Left-Chiral Superfields 277 12.4 Relation Between the Superpotential in
Terms of Superfields and the Superpotential of Chapter 8 285
12.5 The Free Part of the Wess-Zumino Model 288 12.6 Why Does It All Work? 292 12.7 Quiz 295
CHAPTER 13 Supersymmetric Gauge Field Theories in the Superfield Approach 297 13.1 Abelian Gauge Invariance in the
Superfield Formalism 297 13.2 Explicit Interactions Between a Left-Chiral
Multiplet and the Abelian Gauge Multiplet 303 13.3 Lagrangian of a Free Supersymmetric Abelian
Gauge Theory in Superfield Notation 306 13.4 The Abelian Field-Strength Superfield
in Terms of Component Fields 308 13.5 The Free Abelian Supersymmetric Lagrangian
from the Superfield Approach 313 13.6 Supersymmetric QED 317 13.7 Supersymmetric Nonabelian Gauge Theories 322 13.8 Quiz 324
x Supersymmetry Demystified
CHAPTER 14 SUSY Breaking 325 14.1 Spontaneous Supersymmetry Breaking 326 14.2 F-Type SUSY Breaking 330 14.3 The O’Raifeartaigh Model 331 14.4 Mass Spectrum: General Considerations 334 14.5 Mass Spectrum of the O’Raifeartaigh Model
for m 2 ≥ 2g 2M 2 335 14.6 The Supertrace 339 14.7 D-Type SUSY Breaking 340 14.8 Second Example of D-Type SUSY Breaking 343 14.9 Explicit SUSY Breaking 344 14.10 Quiz 350
CHAPTER 15 Introduction to the Minimal Supersymmetric Standard Model 351 15.1 Lightning Review of the Standard Model 351 15.2 Spontaneous Symmetry Breaking in
the Standard Model 360 15.3 Aside on Notation 364 15.4 The Left-Chiral Superfields of the MSSM 365 15.5 The Gauge Vector Superfields 368 15.6 The MSSM Lagrangian 368 15.7 The Superpotential of the MSSM 371 15.8 The General MSSM Superpotential 377 15.9 Quiz 378
CHAPTER 16 Some Phenomenological Implications of the MSSM 379 16.1 Supersymmetry Breaking in the MSSM 380 16.2 The Scalar Potential, Electroweak
Symmetry Breaking, and All That 381 16.3 Finding a Minimum 384 16.4 The Masses of the Gauge Bosons 387 16.5 Masses of the Higgs 392
Contents xi
16.6 Masses of the Leptons, Quarks, and Their Superpartners 399
16.7 Some Other Consequences of the MSSM 400 16.8 Coupling Unification in Supersymmetric GUT 401 16.9 Quiz 412
Final Exam 413
APPENDIX B Solutions to Exercises 427
APPENDIX C Solutions to Quizzes 461
APPENDIX D Solutions to Final Exam 469
Index 475
My heartfelt thanks go to Judy Bass at McGraw-Hill and Preeti Longia Sinha at Glyph International for their guidance, patience, and dedication. I am indebted to Professor Kevin Cahill of the University of New Mexico for many insightful comments on an early draft. A special thanks goes to Professor Pierre Mathieu of Universite Laval for introducing me to supersymmetry when I was an undergraduate and for his continuing support in my research career.
I am deeply grateful to my sisters, Micheline and Maureen, for being there for me throughout the years, through thick and thin. I also want to thank my three rescued cats, Sean, Blue, and Fanny, for keeping me company and keeping me entertained during countless hours of writing and proofreading. Adopt a rescued animal if you can!
I would like to dedicate this book to the memory of my mother, Pierrette, and the three siblings I have lost, Monique, Johnny, and Anne. They are no longer with us in body, but they are still very present in my heart.
Patrick Labelle has a Ph.D. in theoretical physics from Cornell University and held a postdoctoral position at McGill University. He has been teaching physics at the college level for 12 years at Bishop’s University and Champlain Regional College in Sherbrooke, Quebec. Dr. Labelle spent summers doing research both at the Centre Europeen de Recherches Nucleaires (CERN) in Geneva, home of the Large Hadron Collider, and at Fermilab, the world’s second-largest particle accelerator near Chicago. He participated in the French translation of Brian Greene’s The Elegant Universe (in the role of technical adviser).
1.1 What Is Supersymmetry, and Why Is It Exciting (the Short Version)?
The job of a particle theorist is pretty simple. It consists of going through the following four steps:
1. Build a theory that is mathematically self-consistent.
2. Calculate physical processes using the theory.
3. Call experimentalist friends to see if the results of the theory agree with experiments.
4. If they agree, celebrate and hope that the Nobel Committee will notice you. If they don’t, go back to step 1.
That’s pretty much all a particle theorist does with his or her life. However, if one randomly tries every theory that comes to one’s mind, one soon realizes that most of them lead to results that have nothing to do with the real world. It also
2 Supersymmetry Demystified
becomes clear that requiring certain specific properties and symmetries helps to weed out unphysical theories. One such requirement is that the theory must be Lorentz invariant.* Another symmetry that has proved unexpectedly powerful in the construction of the standard model of particle physics is gauge invariance.
However, the standard model is widely considered to be an incomplete theory, even at energies as low as the weak scale (about 100 GeV). One key reason is the so- called hierarchy problem that comes from the fact that the mass of the Higgs particle†
receives large corrections from loop diagrams (see Section 1.3 for a more detailed discussion). These corrections, in principle, can be canceled by a fine-tuning of some parameters of the standard model, but this highly contrived solution seems very unnatural to most physicists (for this reason, the hierarchy problem is also often called a naturalness or fine-tuning problem). There is therefore a strong impetus to build new theories going beyond the standard model (while reproducing all of its successes, of course). For this, as always, it is useful to have some symmetry principle to guide us in designing those new theories. Supersymmetry is such a principle.
The basic idea is fairly simple to state. As the color SU(3) group of the standard model reshuffles the color states of a given quark flavor among themselves and the weak SU(2) group mixes fields appearing in weak doublets (e.g., the left-handed electron and neutrino states), supersymmetry is a symmetry that involves changing bosonic and fermionic fields into one another. Supersymmetry solves the hierar- chy problem in a most ingenious manner: Fermion and boson loop corrections to particle masses cancel one another exactly! Actually, almost all the ultraviolet (high-momentum) divergences of conventional quantum field theory disappear in supersymmetric field theories (i.e., theories invariant under supersymmetry trans- formations), a fact that we will show explicitly with a few examples later in the book.
Unfortunately, exact invariance under supersymmetry implies that each existing particle should have a partner of the same mass and same quantum numbers but with a spin differing by one-half, which is obviously not observed in nature (there is no spin-zero or spin-one particle of the same mass as the electron, for example). However, there is no need to throw away the baby with the bath water. The same type of problem occurs with the standard model, which is based on symmetry under the direct group product SU (3)c × SU (2)L × U (1)Y (a brief review of the standard model will be provided in Chapter 15). We know that the world is not exactly invariant under this symmetry because this would imply that all gauge
* At least at “low energy” relative to the Planck mass, approximately 1019 GeV. At energies comparable to the Planck mass, a theory of quantum gravity is presumably necessary, and then all bets are off.
† The Higgs particle is special in two respects. It is the only fundamental particle of spin zero predicted by the standard model, and it is also the only particle in the standard model that has not been observed yet.
CHAPTER 1 Introduction 3
bosons would be massless, in contradiction with the observed properties of the W ±
and Z bosons. The solution in the standard model is provided by the spontaneous breaking of the symmetry via the Higgs mechanism. One might therefore hope to find some similar trick to spontaneously break supersymmetry. It turns out that the situation in supersymmetry is a bit more challenging than in the standard model, as we will discuss in Chapter 14.
In a different vein, something truly remarkable emerges from the simple idea of mixing bosons and fermions: Performing two successive supersymmetric transfor- mations on a field gives back the same field but evaluated at a different spacetime coordinate than it was initially. Therefore, supersymmetry is intimately linked to spacetime transformations. This is unlike any of the internal symmetries of the stan- dard model. In fact, if we impose invariance of a theory under local supersymmetric transformations, we find ourselves forced to introduce new fields that automatically reproduce Einstein’s general relativity!* The resulting theory is called supergravity.
Supersymmetry has been showing up in some unexpected places. A notable example is superstring theory. When building a string theory that contains both bosonic and fermionic fields, supersymmetry automatically appears. The fact that the theory exhibits invariance under supersymmetry is the reason it is called superstring theory.
Supersymmetry also has been used extensively in many modern developments in mathematical and theoretical physics, such as the Seiberg-Witten duality, the AdS-CFT correspondence, the theory of branes, and so on.
It is important to understand that, for now, supersymmetry is a purely mathemat- ical concept. We do not know if it has a role to play in the description of nature, at least not yet. But it is very likely that we will find out in the very near future. Indeed, the recent opening of the Large Hadron Collider at the Centre Europeen de Recherche Nucleaire (CERN) near Geneva offers the exciting possibility of finally testing whether supersymmetry plays a role in the fundamental laws of physics.
All this makes supersymmetry terribly exciting to learn. Unfortunately, it is quite difficult to teach oneself supersymmetry despite the vast literature on the subject. This brings us to the purpose of this book.
1.2 What This Book Is and What It Is Not There are essentially three roadblocks to teaching oneself supersymmetry. The first is the highly efficient but at the same time horrendously confusing notation that permeates supersymmetry books and articles. It’s hard not to get discouraged by the
* In roughly the same way that making a U (1) global invariance local automatically leads to electromagnetism, for example.
4 Supersymmetry Demystified
dizzying profusion of indices that sometimes do not even seem be used consistently, as in χa , χa , χ a , χa , (σ 2)ab, (σ 2)ab, σ
aa , and so on. There is, of course, a very good reason for introducing all this notation: It helps
immensely in constructing invariant quantities and making expressions as compact as possible. Unfortunately, this also makes it difficult for newcomers to the field because more effort must be put into making sense of the notation than into learning supersymmetry itself! It is possible to keep the notation very simple by not using those…