Subatomic Physics

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Chapter 12 Introduction to Gauge Theories 12.1 Introduction In chapter 7 we introduced both global and local gauge transformations. In this chapter we continue the discussion of gauge invariance and its applications. This invariance has emerged as the primary underpinning of all fundamental subatomic interactions. It is now believed that all forces are described by gauge theories, theories for which local gauge invariance holds. The importance of gauge theories became obvious with the development of the unified electroweak theory; the Standard Model is based on gauge theories for the strong, electromagnetic and weak interactions. In the present chapter we discuss the ideas underlying modern gauge theories. The material is somewhat more difficult than what we have treated so far, but is necessary for understanding the Standard Model. In chapter 7 we saw that additive conservation laws, including charge conservation, follow from a global gauge transformation, Eq. (7.21). We also showed that a local gauge transformation, Eq. (7.27), allows us to identify the charge as the electric one. The development in chapter 7 was for a static charge. However, the Schrödinger equation (7.1) i� ∂ψ ∂t = Hψ (7.1) with the Hamiltonian of Section 10.3, for a particle of charge q under the influence of an electromagnetic field, H = 1 � p − 2m q c A �2 + qA0, (12.1) is also invariant under the combined local gauge transformation, ψ ′ q = e iQɛ(x,t) ψq ≡ UQ(ɛ)ψq, (12.2) 383
384 Introduction to Gauge Theories where Q is the charge operator and or in four-vector notation A ′ 0 = A0 − � ∂ɛ(x,t) ∂t A ′ = A + �c∇ɛ(x,t), (12.3) A ′ µ = Aµ − �c∇µɛ(x,t). (12.4) The local gauge invariance of Maxwell’s equations for classical electricity and magnetism has been known for many decades. In classical electromagnetism, only the electromagnetic fields E and B have physical meaning, and gauge invariance is associated with the partial freedom of choice of the electromagnetic potentials A0 and A of Eqs. (10.37) and (10.38). As we shall show in Section 12.2 the same is not true in quantum mechanics. With the advent of general relativity, which employs local gauge invariance, Weyl in 1919 tried to generalize the electromagnetic local gauge invariance as a geometrical means to unify electromagnetism and gravity. (1) His attempts were unsuccessful and the development lay dormant for over 30 years. However, in the past several decades, local gauge invariance was successfully extended and applied to the unification of electromagnetic and weak interactions. The invariance also underlies the basic theory of all interactions of the Standard Model, grand unified theories, as well as supersymmetric theories that include gravitation. Indeed, all modern descriptions of basic forces are gauge theories. Gauge invariance is a powerful tool. We shall show that it dictates the form of the interaction, and requires massless vector fields, as for instance the electromagnetic field with its massless photon; Table 5.9 shows that the quanta of all subatomic forces have spin 1, and thus correspond to vector fields. In chapter 7, we demonstrated that the form of Eqs. (12.2) and (12.4) leads to invariance under a local gauge transformation. Here we reverse the argument. If the Schrödinger equation (7.1) with H that for a free particle, H = p 2 /2m, isto remain invariant under the local gauge transformation (12.2), then a compensating four-vector field, with time and space components which can be called A0 and A, abbreviated as (A0, A) orsimplyAµ must be introduced. Its concommitant transformations must be given by Eq. (12.4). In the following development we sometimes shall use the shorthand notation for four-vectors. More elaborate manipulations will be shown in brackets and bullets. The requirement for a compensating vector field to maintain invariance of the Schrödinger equation under a local gauge transformation can be seen most easily by introducing the covariant gauge (sometimes simply called covariant derivatives) Dµ =(D0, D) 1 H. Weyl, Ann. Physik 59, 101 (1919).
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SUBAT MIC PHYSICS Third Edition
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SUBAT MIC PHYSICS Ernest M Henley A
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To Elaine and Viviana
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Acknowledgments In writing the pres
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Preface to the First Edition Subato
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Preface to the Third Edition Subato
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General Bibliography The reader of
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Contents Dedication v Acknowledgmen
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Contents xvii 6 Structure of Subato
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Contents xix 12.5 GeneralReferences
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Chapter 1 Background and Language H
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1.2. Units 3 1.2 Units Table 1.1: B
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1.3. Special Relativity, Feynman Di
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1.3. Special Relativity, Feynman Di
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1.4. References 9 Problems 1.1. ∗
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Part I Tools One of the most frustr
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Chapter 2 Accelerators 2.1 Why Acce
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2.1. Why Accelerators? 15 particle
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2.2. Cross Sections and Luminosity
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2.3. Electrostatic Generators (Van
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2.4. Linear Accelerators (Linacs) 2
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2.5. Beam Optics 23 Figure 2.8: Rec
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2.6. Synchrotrons 25 Figure 2.10: E
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2.6. Synchrotrons 27 Photo 3: The p
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2.7. Laboratory and Center-of-Momen
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2.8. Colliding Beams 31 or with Eqs
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2.10. Beam Storage and Cooling 33 l
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2.11. References 35 and in E. Byckl
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2.11. References 37 (b) Extreme rel
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Chapter 3 Passage of Radiation Thro
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3.2. Heavy Charged Particles 41 Fig
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3.2. Heavy Charged Particles 43 −
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3.3. Photons 45 Equation (3.2) also
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3.4. Electrons 47 Figure 3.8: Passa
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3.5. Nuclear Interactions 49 Figure
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3.6. References 51 3.8. A beam of 1
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Chapter 4 Detectors What would a ph
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4.1. Scintillation Counters 55 The
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4.2. Statistical Aspects 57 Or, to
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4.3. Semiconductor Detectors 59 kno
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4.3. Semiconductor Detectors 61 Fig
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4.4. Bubble Chambers 63 Photo 5: Bu
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4.6. Wire Chambers 65 voltage suppl
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4.8. Time Projection Chambers 67 Fi
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4.10. Calorimeters 69 Figure 4.18:
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4.11. Counter Electronics 71 Figure
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4.13. References 73 Table 4.1: Func
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4.13. References 75 4.6. Sketch the
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Part II Particles and Nuclei The si
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Chapter 5 The Subatomic Zoo A conve
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5.1. Mass and Spin. Fermions and Bo
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5.1. Mass and Spin. Fermions and Bo
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5.2. Electric Charge and Magnetic D
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5.3. Mass Measurements 87 Figure 5.
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5.3. Mass Measurements 89 Figure 5.
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5.3. Mass Measurements 91 Earlier,
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5.4. A First Glance at the Subatomi
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5.5. Gauge Bosons 95 Figure 5.13: A
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5.6. Leptons 97 to its momentum, re
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5.7. Decays 99 Figure 5.15: Exponen
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5.7. Decays 101 The function g(ω)
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5.8. Mesons 103 5.8 Mesons Table 5.
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5.9. Baryon Ground States 105 Japan
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5.9. Baryon Ground States 107 neutr
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5.10. Particles and Antiparticles 1
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5.10. Particles and Antiparticles 1
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5.11. Quarks, Gluons, and Intermedi
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5.12. Excited States and Resonances
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5.12. Excited States and Resonances
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5.13. Excited States of Baryons 123
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5.14. References 129 in Section 5.5
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5.14. References 131 5.14. Use the
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5.14. References 133 5.32. ∗ What
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Chapter 6 Structure of Subatomic Pa
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6.2. Rutherford and Mott Scattering
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6.2. Rutherford and Mott Scattering
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6.3. Form Factors 141 The scatterin
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6.4. The Charge Distribution of Sph
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6.4. The Charge Distribution of Sph
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6.5. Leptons Are Point Particles 14
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6.6. Nucleon Elastic Form Factors 1
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6.8. Inelastic Electron and Muon Sc
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6.8. Inelastic Electron and Muon Sc
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6.9. Deep Inelastic Electron Scatte
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6.10. Quark-Parton Model for Deep I
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6.11. More Details on Scattering an
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6.12. References 189 Some character
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6.12. References 191 (c) Show that
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6.12. References 193 6.23. Show tha
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Part III Symmetries and Conservatio
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Chapter 7 Additive Conservation Law
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7.1. Conserved Quantities and Symme
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7.1. Conserved Quantities and Symme
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7.2. The Electric Charge 203 7.2 Th
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7.2. The Electric Charge 205 or [Q,
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7.3. The Baryon Number 207 antineut
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7.4. Lepton and Lepton Flavor Numbe
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7.5. Strangeness Flavor 211 all bel
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7.5. Strangeness Flavor 213 Positiv
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7.6. Additive Quantum Numbers of Qu
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7.7. References 217 There are also
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7.7. References 219 7.12. Follow th
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Chapter 8 Angular Momentum and Isos
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8.2. Symmetry Breaking by a Magneti
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8.4. The Nucleon Isospin 225 8.4 Th
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8.5. Isospin Invariance 227 magneti
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8.6. Isospin of Particles 229 the v
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8.6. Isospin of Particles 231 The a
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8.7. Isospin in Nuclei 233 If the e
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8.8. References 235 Isospin doublet
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8.8. References 237 8.4. Verify the
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Chapter 9 P , C, CP, andT In the pr
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9.1. The Parity Operation 241 P is
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9.2. The Intrinsic Parities of Suba
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9.2. The Intrinsic Parities of Suba
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9.3. Conservation and Breakdown of
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9.4. Charge Conjugation 253 “Is t
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9.4. Charge Conjugation 255 The π
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9.5. Time Reversal 257 if Tψ(t) an
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9.5. Time Reversal 259 found to be
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9.6. The Two-State Problem 261 Hami
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9.7. The Neutral Kaons 263 9.7 The
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9.7. The Neutral Kaons 265 3. The s
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9.7. The Neutral Kaons 267 and only
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9.8. The Fall of CP Invariance 269
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9.9. References 271 • There are t
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9.9. References 273 9.8. * Find inf
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9.9. References 275 9.27. Assume th
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9.9. References 277 9.39. Compare t
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Part IV Interactions In the previou
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Chapter 10 The Electromagnetic Inte
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10.1. The Golden Rule 283 Schrödin
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10.1. The Golden Rule 285 where it
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10.2. Phase Space 287 Figure 10.4:
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10.3. The Classical Electromagnetic
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10.3. The Classical Electromagnetic
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10.4. Photon Emission 293 is a twof
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10.4. Photon Emission 295 the form
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10.4. Photon Emission 297 where V (
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10.5. Multipole Radiation 299 The t
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10.5. Multipole Radiation 301 Figur
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10.6. Electromagnetic Scattering of
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10.6. Electromagnetic Scattering of
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10.7. Vector Mesons as Mediators of
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10.7. Vector Mesons as Mediators of
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10.8. Colliding Beams 311 10.8 Coll
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10.8. Colliding Beams 313 Figure 10
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10.9. Electron-Positron Collisions
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10.10. The Photon-Hadron Interactio
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10.11. Magnetic Monopoles 323 The s
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10.12. References 325 Quantum Elect
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10.12. References 327 (b) Compare t
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10.12. References 329 (a) How would
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Chapter 11 The Weak Interaction Thi
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Page 368 and 369: 11.5. The Muon Decay 347 Linac Pion
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Page 374 and 375: 11.7. Chirality versus Helicity 353
Page 376 and 377: 11.9. Weak Decays of Quarks and the
Page 378 and 379: 11.10. Weak Currents in Nuclear Phy
Page 380 and 381: 11.10. Weak Currents in Nuclear Phy
Page 382 and 383: 11.11. Inverse Beta Decay: Reines a
Page 384 and 385: 11.12. Massive Neutrinos 363 11.12
Page 386 and 387: 11.13. Majorana versus Dirac Neutri
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Page 414 and 415: 12.4. The Higgs Mechanism; Spontane
Page 422 and 423: 12.5. General References 401 12.5 G
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Page 436 and 437: 13.4. Tests of the Standard Model 4
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Page 440 and 441: 13.5. References 419 Physics. 112,
Page 442 and 443: Chapter 14 Strong Interactions All
Page 444 and 445: 14.1. Range and Strength of the Low
Page 446 and 447: 14.2. The Pion-Nucleon Interaction
Page 452 and 453: 14.3. The Form of the Pion-Nucleon
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14.4. The Yukawa Theory of Nuclear
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14.5. Low-Energy Nucleon-Nucleon Fo
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14.6. Meson Theory of the Nucleon-N
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14.7. Strong Processes at High Ener
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14.8. The Standard Model, Quantum C
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14.9. QCD at Low Energies 457 An ex
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14.10. Grand Unified Theories, Supe
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14.11. References 461 and Partons,
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14.11. References 463 Fig. 14.28 14
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14.11. References 465 14.25. The lo
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14.11. References 467 (b) What comb
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Part V Models “A model is like an
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Chapter 15 Quark Models of Mesons a
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15.2. Quarks as Building Blocks of
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15.4. Mesons as Bound Quark States
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15.4. Mesons as Bound Quark States
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15.5. Baryons as Bound Quark States
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15.6. The Hadron Masses 481 Figure
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15.7. QCD and Quark Models of the H
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15.8. Heavy Mesons: Charmonium, Ups
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15.9. Outlook and Problems 493 wher
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15.10. References 495 On a more adv
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15.10. References 497 where J± = J
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15.10. References 499 (b) Apply the
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Chapter 16 Liquid Drop Model, Fermi
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16.1. The Liquid Drop Model 503 Fig
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16.1. The Liquid Drop Model 505 Bey
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16.2. The Fermi Gas Model 507 N = V
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E/A(MeV) 16.3. Heavy Ion Reactions
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16.4. Relativistic Heavy Ion Collis
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16.4. Relativistic Heavy Ion Collis
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16.5. References 515 medium (i.e. s
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16.5. References 517 16.12. Symmetr
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16.5. References 519 16.25. At RHIC
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Chapter 17 The Shell Model The liqu
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17.1. The Magic Numbers 523 The res
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17.2. The Closed Shells 525 solved
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17.2. The Closed Shells 527 Figure
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17.3. The Spin-Orbit Interaction 52
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17.4. The Single-Particle Shell Mod
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17.5. Generalization of the Single-
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17.6. Isobaric Analog Resonances 53
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17.7. Nuclei Far From the Valley of
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17.8. References 539 An elementary
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17.8. References 541 (b) N odd. (c)
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Chapter 18 Collective Model Althoug
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18.1. Nuclear Deformations 545 spin
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18.2. Rotational Spectra of Spinles
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18.2. Rotational Spectra of Spinles
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18.3. Rotational Families 551 A spi
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18.3. Rotational Families 553 2. Th
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18.4. One-Particle Motion in Deform
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18.4. One-Particle Motion in Deform
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18.5. Vibrational States in Spheric
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18.5. Vibrational States in Spheric
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18.6. The Interacting Boson Model 5
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18.7. Highly Excited States; Giant
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18.8. Nuclear Models—Concluding R
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18.8. Nuclear Models—Concluding R
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18.9. References 571 J. S. Vaagen,
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18.9. References 573 18.13. Verify
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18.9. References 575 18.33. Conside
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18.9. References 577 18.51. (a) In
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Chapter 19 Nuclear and Particle Ast
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19.1. The Beginning of the Universe
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19.1. The Beginning of the Universe
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19.2. Primordial Nucleosynthesis 58
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19.3. Stellar Energy and Nucleosynt
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19.4. Stellar Collapse and Neutron
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19.4. Stellar Collapse and Neutron
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19.5. Cosmic Rays 597 We are consta
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19.5. Cosmic Rays 599 Figure 19.11:
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19.6. Neutrino Astronomy and Cosmol
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19.8. References 603 baryon excess
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19.8. References 605 Neutron Stars
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19.8. References 607 (a) What is th
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Index Abundance, 585-586 Accelerato
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Index 611 drift chambers, 66-67 ele
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Index 613 diagrams, 279 electromagn
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Index 615 outlook, 567 Nuclear stru
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Index 617 poles, 494 recurrences, 4
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Index 619 TCP theorem, 269, 448 Tem
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