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Solution Manual for Elementary Particle Physics: An Intuitive Introduction, 1st Edition

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    Solutions Manual for Elementary Particle Physics: An Intuitive Introduction Andrew Larkoski ∗ Physics Department Reed College April 5, 2019 1

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    Contents 1 Introduction 6 1.1 Energy of a Mosquito . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 Yukawa’s Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Mass of the Photon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 Planck Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.5 Expansion of the Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.6 Decay Width of the Z boson . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.7 Decay of Strange Hadrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.8 PDG Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.9 InSpire and arXiv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2 Special Relativity 13 2.1 Properties of Lorentz Transformations . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Rapidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Lorentz-Invariant Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4 Properties of Klein-Gordon Equation . . . . . . . . . . . . . . . . . . . . . . 15 2.5 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.6 Properties of the Clifford Algebra . . . . . . . . . . . . . . . . . . . . . . . . 16 2.7 Relativity of Spin-1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.8 Dark Matter Searches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.9 Top Quark Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3 A Little Group Theory 26 3.1 Representations of the Symmetric Group . . . . . . . . . . . . . . . . . . . . 26 3.2 Lorentz Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.3 Hermitian Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.4 Baker-Campbell-Hausdorff Formula . . . . . . . . . . . . . . . . . . . . . . . 30 3.5 Casimir Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.6 Helicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.7 Symplectic Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.8 π - p Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4 Fermi’s Golden Rule & Feynman Diagrams 34 4.1 Galactic Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.2 Integrating δ -functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.3 Three-Body Phase Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.4 e + e − → e + e − Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.5 Non-Relativistic Limit of Feynman Diagrams . . . . . . . . . . . . . . . . . . 42 4.6 Proton-Proton Total Cross Section . . . . . . . . . . . . . . . . . . . . . . . 44 4.7 Proton Collision Beam at ATLAS . . . . . . . . . . . . . . . . . . . . . . . . 44 4.8 Upper Limits on LUX Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2

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    5 Particle Collider Experiment 48 5.1 Synchrotron Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.2 Limits of the Tracking System . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.3 Reconstructing Muons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.4 Data Quantity from the LHC . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.5 Properties of Poisson Statistics . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.6 Look-Elsewhere Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5.7 Discovery of the Top Quark . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.8 Missing Energy and Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.9 Event Displays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 6 Quantum Electrodynamics in e + e − Collisions 57 6.1 Lorentz Transformation of Spinors . . . . . . . . . . . . . . . . . . . . . . . 57 6.2 Helicity Spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6.3 Spin Analysis of e + e − → μ + μ − . . . . . . . . . . . . . . . . . . . . . . . . . 62 6.4 Spin-0 Photon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 6.5 e + e − → scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 6.6 Decays of the Z boson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 6.7 Inclusive vs. Exclusive Cross Sections . . . . . . . . . . . . . . . . . . . . . . 65 6.8 Finite Decay Width Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 7 Quarks & Gluons 68 7.1 Plus-Function Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 7.2 Breit Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 7.3 Form Factor Evolution Equation . . . . . . . . . . . . . . . . . . . . . . . . . 72 7.4 Infrared and Collinear Safety . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 7.5 Properties of Helicity Spinors . . . . . . . . . . . . . . . . . . . . . . . . . . 75 7.6 More Helicity Spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 7.7 The Drell-Yan Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 7.8 Thrust in Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 8 Quantum Chromodynamics 81 8.1 Masslessness of the Gluon . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 8.2 Bianchi Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 8.3 Instantons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 8.4 Wilson Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 8.5 su (2) Lie Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 8.6 Casimir Invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 8.7 Adjoint Representation of SU(3) . . . . . . . . . . . . . . . . . . . . . . . . . 91 8.8 Running Couplings of QED and QCD . . . . . . . . . . . . . . . . . . . . . . 92 3

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    9 Parton Evolution and Jets 95 9.1 Dilation Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 9.2 Expansion of Differential Cross Section for Thrust . . . . . . . . . . . . . . . 96 9.3 Jet Multiplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 9.4 Properties of the DGLAP Equation . . . . . . . . . . . . . . . . . . . . . . . 100 9.5 Resummation of Q 2 with DGLAP . . . . . . . . . . . . . . . . . . . . . . . . 100 9.6 Jet Mass at the LHC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 9.7 Underlying Event at Hadron Colliders . . . . . . . . . . . . . . . . . . . . . . 104 9.8 Jet Event Display . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 10 Parity Violation 107 10.1 Time Reversal of Spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 10.2 Charge Conjugation of Spinors . . . . . . . . . . . . . . . . . . . . . . . . . . 108 10.3 CPT on Spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 10.4 C, P, T in Electromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 10.5 Electron Spin in Muon Decay . . . . . . . . . . . . . . . . . . . . . . . . . . 111 10.6 Endpoint of Electron Energy in Muon Decays . . . . . . . . . . . . . . . . . . 112 10.7 Kinematics of the IceCube Experiment . . . . . . . . . . . . . . . . . . . . . 114 10.8 High-Energy Neutrino Cross Sections . . . . . . . . . . . . . . . . . . . . . . 116 11 The Mass Scales of the Weak Force 122 11.1 Maxwell with a Massive Photon . . . . . . . . . . . . . . . . . . . . . . . . . 122 11.2 Scalar Higgs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 11.3 Charge 0 Higgs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 11.4 Forms of SU(2) Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 11.5 Unification of Couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 11.6 When V − A and When Electroweak? . . . . . . . . . . . . . . . . . . . . . . 126 11.7 Charged Current DIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 11.8 Left-Handed Coupling of the W Boson? . . . . . . . . . . . . . . . . . . . . . 131 12 Consequences of the Weak Interactions 134 12.1 Mass and Flavor Basis Commutator . . . . . . . . . . . . . . . . . . . . . . . 134 12.2 Unitarity of the CKM Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 137 12.3 Jarlskog Invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 12.4 Extra Quark Generations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 12.5 Measuring the Cabibbo Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 12.6 Non-Relativistic Limit of Neutrino Oscillations . . . . . . . . . . . . . . . . . 141 12.7 Neutrinos for Nuclear Non-Proliferation . . . . . . . . . . . . . . . . . . . . 142 12.8 Neutrinos from SN 1987a . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 12.9 Solar Neutrino Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 4

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    13 The Higgs Boson 148 13.1 W Boson Decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 13.2 pp → W + W − Backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 13.3 Searching for H → W + W − . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 13.4 H → γγ Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 13.5 Higgs Production Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 13.6 Landau-Yang Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 13.7 Combining Uncorrelated Measurements . . . . . . . . . . . . . . . . . . . . . 155 13.8 Testing the Spin-2 Higgs Boson Hypothesis . . . . . . . . . . . . . . . . . . . 156 14 Particle Physics at the Frontier 157 14.1 Neutrinoless Double- β Decay at EXO . . . . . . . . . . . . . . . . . . . . . . 157 14.2 Double-Higgs Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 14.3 Borel Summation of a Convergent Series . . . . . . . . . . . . . . . . . . . . 158 14.4 The International Linear Collider . . . . . . . . . . . . . . . . . . . . . . . . 158 14.5 The Largest Possible Collider . . . . . . . . . . . . . . . . . . . . . . . . . . 159 5

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    1 Introduction 1.1 Energy of a Mosquito The mass of a mosquito is approximately m = 2 . 5 × 10 − 6 kg and it flies at approximately v = 0 . 1 m/s, or so. Therefore, its kinetic energy is K = 1 2 mv 2 = 1 . 25 × 10 − 8 J . (1) One electron volt is about 1 . 6 × 10 − 19 J, so the energy in eV of a flying mosquito is K = 1 . 25 × 10 − 8 1 . 6 × 10 − 19 eV ' 7 . 8 × 10 10 eV . (2) The energy per nucleon of the flying mosquito can be found by dividing the total energy found above by the number of protons and neutrons in the mosquito. With a total mass of 2 . 5 × 10 − 6 kg and the mass of the proton/neutron is approximately m p = 1 . 67 × 10 − 27 kg, the total number of nucleons in the mosquito are N n ' 2 . 5 × 10 − 6 1 . 67 × 10 − 27 ' 1 . 5 × 10 21 . (3) Therefore the kinetic energy per nucleon of the mosquito is about K N n ' 7 . 8 × 10 10 1 . 5 × 10 21 eV ' 5 . 2 × 10 − 11 eV . (4) This is about 23 orders of magnitude smaller than the energy of protons at the LHC! 1.2 Yukawa’s Theory The radius of an atomic nucleus is on the order of a femtometer, 10 − 15 m. To turn this into a mass or energy, we divide the product ℏ c by this distance. This is E = ℏ c x = (1 . 05 × 10 − 34 ) · (3 × 10 8 ) 10 − 15 J ' 3 × 10 − 11 J . (5) To convert to eV, we divide by the ratio eV/J ' 1 . 6 × 10 − 19 , so that E = 3 × 10 − 11 1 . 6 × 10 − 19 ' 2 × 10 8 eV = 200 MeV . (6) That is, the pion has a mass of about 200 MeV. 6

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    1.3 Mass of the Photon If Maxwell’s equations describe the magnetic field of the Milky Way galaxy, this sets an upper bound on the mass of the photon. The diameter of the Milky Way is about 100,000 light-years, which in meters is approximately 100 , 000 l-y = 10 5 · ( 3 × 10 8 ) · ( π × 10 7 ) ' 10 21 m . (7) In this expression, we used the fact that, to better than 1% accuracy, the number of seconds in a year is π × 10 7 . If electromagnetism as we understand it describes the galactic magnetic field at this distance, the photon must be able to have a wavelength that is at least this size. The corresponding upper bound on the minimum photon energy is E < ℏ c x ' (1 . 05 × 10 − 34 ) · (3 × 10 8 ) 10 21 J ' 3 × 10 − 47 J . (8) In electron volts, this corresponds to E < 3 × 10 − 47 1 . 6 × 10 − 19 eV ' 2 × 10 − 28 eV . (9) So, the mass of the photon must be less than about 10 − 28 eV. Converting this to kg, we divide the energy in Joules by c 2 : m < 3 × 10 − 47 (3 × 10 8 ) 2 ' 3 × 10 − 64 kg . (10) The mass of the electron is about 1/2 MeV, so this is about 34 orders of magnitude smaller. While this limit is extremely impressive, the assumptions necessary to describe the galac- tic magnetic field and connect it to Maxwell’s equations in particular are a bit tenuous, so this result is not used by the PDG to set a limit on the photon mass. 1.4 Planck Units 1.4 (a) The Planck time, t P , can be expressed as a product of Newton’s constant G N , ℏ , and the speed of light c raised to some powers: t P = G α N ℏ β c γ , (11) where α , β , and γ are some numerical powers. We can find the powers by matching units on both sides of the expression. c is a velocity and so [ c ] = LT − 1 , (12) where L is a length and T is a time unit. ℏ has units of energy times time or that [ ℏ ] = M L 2 T − 1 , (13) 7

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    where M is a mass unit. Finally, the units of G N can be determined by Newton’s law of gravitation and Newton’s second law: ~ F g = − G N m 1 m 2 r 2 ˆ r = m 1 d 2 ~ r dt 2 , (14) from which it follows that [ G N ] = M − 1 L 3 T − 2 . (15) Plugging these into the expression for the Planck time t P we have [ t P ] = T = [ G N ] α [ ℏ ] β [ c ] γ = M − α L 3 α T − 2 α M β L 2 β T − β L γ T − γ . (16) Demanding that there is no mass unit requires that − α + β = 0 , (17) or that α = β . Demanding that there be no length unit requires 3 α + 2 β + γ = 0 = 5 α + γ = 0 , (18) or that γ = − 5 α . Finally, demanding that there be one unit of time requires that − 2 α − β − γ = 1 = 2 α , (19) or that α = 1 / 2. It then follows that the Planck time is t P = √ G N ℏ c 5 . (20) The value of Newton’s constant in SI units is G N = 6 . 67 × 10 − 11 kg − 1 m 3 s − 2 . It then follows that the Planck time is t P = √ (6 . 67 × 10 − 11 ) · (1 . 05 × 10 − 34 ) (3 × 10 8 ) 5 s ' 5 . 4 × 10 − 44 s , (21) which is pretty small! 1.4 (b) Now, we’re asked to find the Planck mass, m P . Because we already have the Planck time, t P , we can find the Planck mass pretty easily. Note that the quantity E P = ℏ t P , (22) is an energy. Then, the Planck mass is found by dividing by c 2 : m P = ℏ t P c 2 = √ ℏ c G N ' 2 × 10 − 8 kg . (23) 8

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    Expressed in eV, the Planck mass is m P c 2 1 . 6 × 10 − 19 ' 1 . 2 × 10 28 eV . (24) The proton mass is about a GeV, or 10 9 eV, so the Planck mass is about 19 orders of magnitude larger! 1.4 (c) For two particles of mass m 1 and m 2 and electric charges q 1 and q 2 , the ratio of the electric force ~ F E to the gravitational force ~ F g between them is | ~ F E | | ~ F g | = 1 4 π 0 G N | q 1 || q 2 | m 1 m 2 , (25) where  0 is the permittivity of free space. The proton and electron both have an electric charge magnitude of the fundamental unit of charge e = 1 . 6 × 10 − 19 , so plugging in numbers, the ratio of forces is | ~ F E | | ~ F g | ' 1 4 π (8 . 85 × 10 − 12 ) (6 . 67 × 10 − 11 ) (1 . 6 × 10 − 19 ) 2 (1 . 67 × 10 − 27 ) (9 . 11 × 10 − 31 ) ' 2 . 2 × 10 39 . (26) 1.5 Expansion of the Universe 1.5 (a) The CMB has a temperature of 2.7 K. With Boltzmann’s constant k B , we can turn this into a corresponding energy. We find E CMB = k B T = 2 . 7 × 1 . 38 × 10 − 23 J ' 3 . 7 × 10 − 23 J . (27) To determine the energy in eV, we divide by 1 . 6 × 10 − 19 : E CMB ' 3 . 7 × 10 − 23 1 . 6 × 10 − 19 eV ' 2 . 3 × 10 − 4 eV . (28) 1.5 (b) The ground state energy of hydrogen is E g = − 13 . 6 eV. So, when the temperature of the universe was less than the energy of 13.6 eV, electrons and protons could become bound and form hydrogen. To determine this temperature, we work backward from the steps of the first part of this problem, multiplying by the factor 1 . 6 × 10 − 19 and then dividing by k B . We find T recomb. = 1 . 6 × 10 − 19 | E g | k B K ' 1 . 6 × 10 5 K . (29) 9

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    1.5 (c) We want to calculate the ratio of the wavelength of CMB photons observed today, λ today , by the wavelength at recombination, λ recomb. . This ratio is λ today λ recomb. = f recomb. f today = E recomb. E today = T recomb. T today . (30) In this chain of equalities, we used the fact that wavelength λ is inversely proportional to frequency f and the frequency of light is proportional to its energy. From what was developed in the previous parts, the energy of the photons is proportional to its temperature. So, the redshift factor is just the ratio of the temperature at recombination to the temperature today: λ today λ recomb. = T recomb. T today ' 1 . 6 × 10 5 2 . 7 ' 5 . 9 × 10 4 . (31) As mentioned in the problem, this is a factor of about 30 larger than the true result when thermodynamics are properly taken into account. 1.6 Decay Width of the Z boson 1.6 (a) From the plot of Fig. 1.5, the maximum value of the peak of the distribution is about 32 nb. Therefore, half of this is 16 nb. The lower point of the distribution at which it takes a value of 16 nb is at approximately 90 GeV, while the higher point is at about 92.5 GeV. Therefore, the width, or full-width at half-maximum is the difference of these two values, or 2.5 GeV. 1.6 (b) To determine the lifetime in seconds of the Z boson from its width, we need to relate the width to a time through the energy-time uncertainty relation: ∆ t ' ℏ ∆ E . (32) To convert the width from natural units to SI, we need to multiply by the factor of 1 . 6 × 10 − 19 so that the lifetime in seconds is ∆ t ' ℏ ∆ E ' 1 . 05 × 10 − 34 (2 . 5 × 10 9 ) · (1 . 6 × 10 − 19 ) ' 2 . 6 × 10 − 25 s . (33) We also needed to include a factor of 10 9 to account for the fact that the width is 2 . 5 GeV = 2 . 5 × 10 9 eV. 10

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    1.6 (c) Through the energy-time uncertainty principle, lifetime and decay width are inversely pro- portional. Therefore, if the width approaches 0, then the lifetime diverges: the particle lives forever. Correspondingly, if the width gets very large, then the lifetime approaches 0: the particle decays instantly. 1.7 Decay of Strange Hadrons The Ω − hadron travels about 3 cm = 0 . 03 m before decaying which corresponds to a lifetime by multiplication by its velocity. While its velocity is not known, it will be an appreciable fraction of the speed of light c , so we can just assume that it is c . Dividing by c , the lifetime τ is τ ' 0 . 03 3 × 10 8 s ' 10 − 10 s . (34) To convert to the decay width ∆ E , we use the energy-time uncertainty relationship, and the factor 1 . 6 × 10 − 19 to convert to eV. We then find ∆ E = ℏ τ ' 1 . 05 × 10 − 34 10 − 10 · (1 . 6 × 10 − 19 ) eV ' 6 . 6 × 10 − 5 eV . (35) 1.8 PDG Review 1.8 (a) From the PDG, the lower bound on the lifetime of the proton is 2 . 1 × 10 29 years. To have a reasonable probability to observe one proton decay in a year, you would need at least 2 . 1 × 10 29 protons, if the lifetime were exactly at the lower bound. Water, H 2 O, consists of 10 protons (and 8 neutrons), so we would need to observe 2 . 1 × 10 28 water molecules for a year. The volume of this amount of water can be found by first identifying the number of moles of water: 2 . 1 × 10 28 6 . 02 × 10 23 mol ' 3 . 5 × 10 4 mol . (36) Because water consists of 18 total protons and neutrons, this amount of water has a mass of ( 3 . 5 × 10 4 ) · 18 g ' 630 kg . (37) Water has a density of 1000 kg/m 3 , so the total volume of water needed is about 630 1000 m 3 = 0 . 63 m 3 . (38) This is roughly the volume of a large bathtub. 11

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    1.8 (b) From the “Particle Listings” section of the PDG, the masses of the W , Z , and Higgs bosons and the top quark are: m W = 80 . 379 GeV , (39) m Z = 91 . 1876 GeV , (40) m H = 125 . 18 GeV , (41) m t = 173 . 0 GeV . (42) The mass of the proton or neutron is approximately 1 GeV (actually slightly less), and so the value of the mass in GeV can be approximately used to identify the atomic mass of elements with about the same mass. For example, Krypton has an atomic mass of about 83, which is close to the mass of the W boson. Zirconium has an atomic mass of about 91, close to the mass of the Z boson. Tellurium has an atomic mass of about 127, close to the mass of the Higgs boson. Finally, Ytterbium has an atomic mass of about 174, close to the mass of the top quark. Students may find a slightly different selection of elements from a more precise accounting of the proton and neutron masses or identification of different isotopes. 1.8 (c) We want to identify the masses of the particles involved in the bubble chamber trace of Fig. 1.6. Again, we use the “Particle Listings” section of the PDG, and we have to do a bit of sleuthing to identify all the particles by their symbols. Their masses are: m K − = 493 . 677 MeV , (43) m Ω − = 1672 . 43 MeV , (44) m K 0 = 497 . 611 MeV , (45) m π − = 139 . 57061 MeV , (46) m Ξ 0 = 1314 . 82 MeV , (47) m K + = 493 . 677 MeV , (48) m Λ 0 = 1115 . 683 MeV , (49) m p = 938 . 2720813 MeV . (50) The width, or lifetime, from the PDG of the Ω − baryon is 0 . 821 × 10 − 10 s, which is very close to our very simple estimate! 1.9 InSpire and arXiv 1.9 (a) At InSpire, we can search for Noether’s papers and find her most highly-cited paper. In fact, InSpire only has one of her papers listed (all others are pure mathematics), which is 12

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    “Invariant Variation Problems.” 1.9 (b) We can also search by date. To find all papers from 1967, we use the command “find date = 1967”. Then, we can sort by decreasing order in citation count. The two most highly-cited papers are: S. Weinberg, “A Model of Leptons,” Phys. Rev. Lett. 19 , 1264 (1967). A. D. Sakharov, “Violation of CP Invariance, C asymmetry, and baryon asym- metry of the universe,” Pisma Zh. Eksp. Teor. Fiz. 5 , 32 (1967) [JETP Lett. 5 , 24 (1967)] [Sov. Phys. Usp. 34 , no. 5, 392 (1991)] [Usp. Fiz. Nauk 161 , no. 5, 61 (1991)]. 2 Special Relativity 2.1 Properties of Lorentz Transformations We’re asked to verify that the matrix Λ ν μ =     γ 0 0 γβ 0 1 0 0 0 0 1 0 γβ 0 0 γ     (51) leaves the metric invariant: Λ ᵀ η Λ = η . (52) First, note that the matrix Λ is symmetric: Λ ᵀ = Λ. So, multiplying from the left, we have Λ ᵀ η =     γ 0 0 γβ 0 1 0 0 0 0 1 0 γβ 0 0 γ         1 0 0 0 0 − 1 0 0 0 0 − 1 0 0 0 0 − 1     =     γ 0 0 − γβ 0 − 1 0 0 0 0 − 1 0 γβ 0 0 − γ     . (53) Continuing, multiplying by Λ on the right produces Λ ᵀ η Λ =     γ 0 0 − γβ 0 − 1 0 0 0 0 − 1 0 γβ 0 0 − γ         γ 0 0 γβ 0 1 0 0 0 0 1 0 γβ 0 0 γ     (54) =     γ 2 (1 − β 2 ) 0 0 0 0 − 1 0 0 0 0 − 1 0 0 0 0 − γ 2 (1 − β 2 )     = η . Recall that 1 − β 2 = γ − 2 , from which the result follows. 13

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    2.2 Rapidity The rapidity y is defined to be y = 1 2 log E + p z E − p z . (55) We can perform a Lorentz boost along the ˆ z axis by a velocity β by multiplication of the momentum four-vector by a Lorentz matrix:     γ 0 0 γβ 0 1 0 0 0 0 1 0 γβ 0 0 γ         E 0 0 p z     =     γ ( E + βp z ) 0 0 γ ( p z + βE )     . (56) Therefore, under a Lorentz transformation, the rapidity transforms as y → 1 2 log γ ( E + βp z ) + γ ( p z + βE ) γ ( E + βp z ) − γ ( p z + βE ) = 1 2 log (1 + β )( E + p z ) (1 − β )( E − p z ) = y + 1 2 log 1 + β 1 − β . (57) That is, under a Lorentz boost along the ˆ z axis, the rapidity transforms additively. 2.3 Lorentz-Invariant Measure Under a Lorentz transformation, the coordinate four-vector x μ transforms to x ′ μ = Λ μ ν x ν . (58) Then, under a Lorentz transformation, the coordinate measure d 4 x transforms as d 4 x ′ = | J | d 4 x , (59) where J is the Jacobian formed from the determinant of the derivative matrix: J = det ∂x ′ μ ∂x ν . (60) From the Lorentz transformation above in terms of the matrix Λ, this partial derivative is ∂x ′ μ ∂x ν = Λ μ ν . (61) Therefore, the Jacobian is just the determinant of the Lorentz-transformation matrix: J = det Λ . (62) By the definition of Λ, it satisfies Λ ᵀ η Λ = η , (63) 14

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    which enables us to calculate the determinant of Λ. First, note that transposition doesn’t change the determinant: det Λ = det Λ ᵀ . (64) Also, the determinant of the metric η is just the product of its non-zero elements: det η = − 1 . (65) Then, taking the determinant of the Lorentz transformation equation, we have det(Λ ᵀ η Λ) = − (det Λ) 2 = − 1 . (66) Therefore, | det Λ | = 1 = J , and so after Lorentz transformation the measure d 4 x is un- changed. 2.4 Properties of Klein-Gordon Equation The solution of the Klein-Gordon equation is an exponential phase function, φ ( x ) = e − ip · x = e − i ( Et − ~ p · ~ x ) . (67) We can find the frequency by adding the period T to the time t and demanding that the field φ ( x ) is unchanged: e − i ( E ( t + T ) − ~ p · ~ x ) = e − i ( Et − ~ p · ~ x ) , (68) or that exp[ − iET ] = 1. Then, the period T = 2 π/E and so the frequency f = E/ (2 π ). A similar procedure can be used to determine the wavelength of the solution. The wavelength λ is then λ = 2 π | ~ p | . (69) It then follows that the phase velocity is the just the product of the frequency and wavelength: v = λf = E | ~ p | . (70) 2.5 Maxwell’s Equations Gauss’s law has already been identified as the 0 th component of the equations of motion of ∂ μ F μν = J ν . (71) Now, let’s take ν = i , a spatial coordinate. Then, J i is the i th component of the current vector and the left side of this equation is ∂ μ F μi = − ∂ 0 F 0 i + ∂ i F ii + ∂ j F ji + ∂ k F ki = − ∂E i ∂t + ( ∂B k ∂j − ∂B j ∂k ) . (72) 15

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