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

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Solutions Manual forElementary Particle Physics:An Intuitive IntroductionAndrew Larkoski∗Physics DepartmentReed CollegeApril 5, 20191

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Contents1Introduction61.1Energy of a Mosquito. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .61.2Yukawa’s Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .61.3Mass of the Photon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .71.4Planck Units. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .71.5Expansion of the Universe. . . . . . . . . . . . . . . . . . . . . . . . . . . .91.6Decay Width of the Z boson. . . . . . . . . . . . . . . . . . . . . . . . . . .101.7Decay of Strange Hadrons. . . . . . . . . . . . . . . . . . . . . . . . . . . .111.8PDG Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .111.9InSpire and arXiv. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .122Special Relativity132.1Properties of Lorentz Transformations. . . . . . . . . . . . . . . . . . . . . .132.2Rapidity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .142.3Lorentz-Invariant Measure. . . . . . . . . . . . . . . . . . . . . . . . . . . .142.4Properties of Klein-Gordon Equation. . . . . . . . . . . . . . . . . . . . . .152.5Maxwell’s Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .152.6Properties of the Clifford Algebra. . . . . . . . . . . . . . . . . . . . . . . .162.7Relativity of Spin-1/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .182.8Dark Matter Searches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .212.9Top Quark Decay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .243A Little Group Theory263.1Representations of the Symmetric Group. . . . . . . . . . . . . . . . . . . .263.2Lorentz Group. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .283.3Hermitian Matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .293.4Baker-Campbell-Hausdorff Formula. . . . . . . . . . . . . . . . . . . . . . .303.5Casimir Operator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .303.6Helicity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .313.7Symplectic Group. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .323.8π-pScattering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .334Fermi’s Golden Rule & Feynman Diagrams344.1Galactic Collisions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .344.2Integratingδ-functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .374.3Three-Body Phase Space. . . . . . . . . . . . . . . . . . . . . . . . . . . . .384.4e+e−→e+e−Scattering. . . . . . . . . . . . . . . . . . . . . . . . . . . . .414.5Non-Relativistic Limit of Feynman Diagrams. . . . . . . . . . . . . . . . . .424.6Proton-Proton Total Cross Section. . . . . . . . . . . . . . . . . . . . . . .444.7Proton Collision Beam at ATLAS. . . . . . . . . . . . . . . . . . . . . . . .444.8Upper Limits on LUX Bounds. . . . . . . . . . . . . . . . . . . . . . . . . .462

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5Particle Collider Experiment485.1Synchrotron Losses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .485.2Limits of the Tracking System. . . . . . . . . . . . . . . . . . . . . . . . . .495.3Reconstructing Muons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .515.4Data Quantity from the LHC. . . . . . . . . . . . . . . . . . . . . . . . . .515.5Properties of Poisson Statistics. . . . . . . . . . . . . . . . . . . . . . . . .515.6Look-Elsewhere Effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .535.7Discovery of the Top Quark. . . . . . . . . . . . . . . . . . . . . . . . . . .555.8Missing Energy and Neutrinos. . . . . . . . . . . . . . . . . . . . . . . . . .555.9Event Displays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .566Quantum Electrodynamics ine+e−Collisions576.1Lorentz Transformation of Spinors. . . . . . . . . . . . . . . . . . . . . . .576.2Helicity Spinors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .606.3Spin Analysis ofe+e−→μ+μ−. . . . . . . . . . . . . . . . . . . . . . . . .626.4Spin-0 Photon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .626.5e+e−→scalars. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .626.6Decays of the Z boson. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .646.7Inclusive vs. Exclusive Cross Sections. . . . . . . . . . . . . . . . . . . . . .656.8Finite Decay Width Effects. . . . . . . . . . . . . . . . . . . . . . . . . . . .667Quarks & Gluons687.1Plus-Function Expansion. . . . . . . . . . . . . . . . . . . . . . . . . . . . .687.2Breit Frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .707.3Form Factor Evolution Equation. . . . . . . . . . . . . . . . . . . . . . . . .727.4Infrared and Collinear Safety. . . . . . . . . . . . . . . . . . . . . . . . . . .737.5Properties of Helicity Spinors. . . . . . . . . . . . . . . . . . . . . . . . . .757.6More Helicity Spinors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .767.7The Drell-Yan Process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .787.8Thrust in Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .818Quantum Chromodynamics818.1Masslessness of the Gluon. . . . . . . . . . . . . . . . . . . . . . . . . . . .818.2Bianchi Identity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .828.3Instantons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .848.4Wilson Lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .868.5su(2)Lie Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .888.6Casimir Invariant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .898.7Adjoint Representation of SU(3). . . . . . . . . . . . . . . . . . . . . . . . .918.8Running Couplings of QED and QCD. . . . . . . . . . . . . . . . . . . . . .923

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9Parton Evolution and Jets959.1Dilation Operator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .959.2Expansion of Differential Cross Section for Thrust. . . . . . . . . . . . . . .969.3Jet Multiplicity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .979.4Properties of the DGLAP Equation. . . . . . . . . . . . . . . . . . . . . . .1009.5Resummation of Q2with DGLAP. . . . . . . . . . . . . . . . . . . . . . . .1009.6Jet Mass at the LHC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1029.7Underlying Event at Hadron Colliders. . . . . . . . . . . . . . . . . . . . . .1049.8Jet Event Display. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10710 Parity Violation10710.1Time Reversal of Spinors. . . . . . . . . . . . . . . . . . . . . . . . . . . . .10710.2Charge Conjugation of Spinors. . . . . . . . . . . . . . . . . . . . . . . . . .10810.3CPT on Spinors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10910.4C, P, T in Electromagnetism. . . . . . . . . . . . . . . . . . . . . . . . . . .11010.5Electron Spin in Muon Decay. . . . . . . . . . . . . . . . . . . . . . . . . .11110.6Endpoint of Electron Energy in Muon Decays. . . . . . . . . . . . . . . . . .11210.7Kinematics of the IceCube Experiment. . . . . . . . . . . . . . . . . . . . .11410.8High-Energy Neutrino Cross Sections. . . . . . . . . . . . . . . . . . . . . .11611 The Mass Scales of the Weak Force12211.1Maxwell with a Massive Photon. . . . . . . . . . . . . . . . . . . . . . . . .12211.2Scalar Higgs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12311.3Charge 0 Higgs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12311.4Forms of SU(2) Matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . .12311.5Unification of Couplings. . . . . . . . . . . . . . . . . . . . . . . . . . . . .12411.6WhenV−Aand When Electroweak?. . . . . . . . . . . . . . . . . . . . . .12611.7Charged Current DIS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12811.8Left-Handed Coupling of the W Boson?. . . . . . . . . . . . . . . . . . . . .13112 Consequences of the Weak Interactions13412.1Mass and Flavor Basis Commutator. . . . . . . . . . . . . . . . . . . . . . .13412.2Unitarity of the CKM Matrix. . . . . . . . . . . . . . . . . . . . . . . . . .13712.3Jarlskog Invariant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13812.4Extra Quark Generations. . . . . . . . . . . . . . . . . . . . . . . . . . . . .14012.5Measuring the Cabibbo Angle. . . . . . . . . . . . . . . . . . . . . . . . . . .14112.6Non-Relativistic Limit of Neutrino Oscillations. . . . . . . . . . . . . . . . .14112.7Neutrinos for Nuclear Non-Proliferation. . . . . . . . . . . . . . . . . . . .14212.8Neutrinos from SN 1987a. . . . . . . . . . . . . . . . . . . . . . . . . . . .14512.9Solar Neutrino Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . .1484

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13 The Higgs Boson14813.1W Boson Decays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14813.2pp→W+W−Backgrounds. . . . . . . . . . . . . . . . . . . . . . . . . . . .14913.3Searching forH→W+W−. . . . . . . . . . . . . . . . . . . . . . . . . . . .14913.4H→γγRate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15013.5Higgs Production Rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15113.6Landau-Yang Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15313.7Combining Uncorrelated Measurements. . . . . . . . . . . . . . . . . . . . .15513.8Testing the Spin-2 Higgs Boson Hypothesis. . . . . . . . . . . . . . . . . . .15614 Particle Physics at the Frontier15714.1Neutrinoless Double-βDecay at EXO. . . . . . . . . . . . . . . . . . . . . .15714.2Double-Higgs Production. . . . . . . . . . . . . . . . . . . . . . . . . . . . .15714.3Borel Summation of a Convergent Series. . . . . . . . . . . . . . . . . . . .15814.4The International Linear Collider. . . . . . . . . . . . . . . . . . . . . . . .15814.5The Largest Possible Collider. . . . . . . . . . . . . . . . . . . . . . . . . .1595

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1Introduction1.1Energy of a MosquitoThe mass of a mosquito is approximatelym= 2.5×10−6kg and it flies at approximatelyv= 0.1 m/s, or so. Therefore, its kinetic energy isK = 12mv2= 1.25×10−8J.(1)One electron volt is about 1.6×10−19J, so the energy in eV of a flying mosquito isK = 1.25×10−81.6×10−19eV'7.8×1010eV.(2)The energy per nucleon of the flying mosquito can be found by dividing the total energyfound above by the number of protons and neutrons in the mosquito. With a total mass of2.5×10−6kg and the mass of the proton/neutron is approximatelymp= 1.67×10−27kg,the total number of nucleons in the mosquito areNn'2.5×10−61.67×10−27'1.5×1021.(3)Therefore the kinetic energy per nucleon of the mosquito is aboutKNn'7.8×10101.5×1021eV'5.2×10−11eV.(4)This is about 23 orders of magnitude smaller than the energy of protons at the LHC!1.2Yukawa’s TheoryThe radius of an atomic nucleus is on the order of a femtometer, 10−15m. To turn this intoa mass or energy, we divide the productℏcby this distance. This isE=ℏcx= (1.05×10−34)·(3×108)10−15J'3×10−11J.(5)To convert to eV, we divide by the ratio eV/J'1.6×10−19, so thatE=3×10−111.6×10−19'2×108eV = 200 MeV.(6)That is, the pion has a mass of about 200 MeV.6

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1.3Mass of the PhotonIf Maxwell’s equations describe the magnetic field of the Milky Way galaxy, this sets anupper bound on the mass of the photon. The diameter of the Milky Way is about 100,000light-years, which in meters is approximately100,000 l-y = 105·(3×108)·(π×107)'1021m.(7)In this expression, we used the fact that, to better than 1% accuracy, the number of secondsin a year isπ×107. If electromagnetism as we understand it describes the galactic magneticfield 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 isE <ℏcx'(1.05×10−34)·(3×108)1021J'3×10−47J.(8)In electron volts, this corresponds toE <3×10−471.6×10−19eV'2×10−28eV.(9)So, the mass of the photon must be less than about 10−28eV. Converting this to kg, wedivide the energy in Joules byc2:m <3×10−47(3×108)2'3×10−64kg.(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, sothis result is not used by the PDG to set a limit on the photon mass.1.4Planck Units1.4 (a)The Planck time,tP, can be expressed as a product of Newton’s constantGN,ℏ, and thespeed of lightcraised to some powers:tP=GαNℏβcγ,(11)whereα,β, andγare some numerical powers. We can find the powers by matching unitson both sides of the expression.cis a velocity and so[c] =LT−1,(12)whereLis a length andTis a time unit.ℏhas units of energy times time or that[ℏ] =M L2T−1,(13)7

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whereMis a mass unit.Finally, the units ofGNcan be determined by Newton’s law ofgravitation and Newton’s second law:~Fg=−GNm1m2r2ˆr=m1d2~rdt2,(14)from which it follows that[GN] =M−1L3T−2.(15)Plugging these into the expression for the Planck timetPwe have[tP] =T= [GN]α[ℏ]β[c]γ=M−αL3αT−2αMβL2βT−βLγT−γ.(16)Demanding that there is no mass unit requires that−α+β= 0,(17)or thatα=β. Demanding that there be no length unit requires3α+ 2β+γ= 0 = 5α+γ= 0,(18)or thatγ=−5α. Finally, demanding that there be one unit of time requires that−2α−β−γ= 1 = 2α ,(19)or thatα= 1/2. It then follows that the Planck time istP=√GNℏc5.(20)The value of Newton’s constant in SI units isGN= 6.67×10−11kg−1m3s−2.It thenfollows that the Planck time istP=√(6.67×10−11)·(1.05×10−34)(3×108)5s'5.4×10−44s,(21)which is pretty small!1.4 (b)Now, we’re asked to find the Planck mass,mP. Because we already have the Planck time,tP, we can find the Planck mass pretty easily. Note that the quantityEP=ℏtP,(22)is an energy. Then, the Planck mass is found by dividing byc2:mP=ℏtPc2=√ℏcGN'2×10−8kg.(23)8

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Expressed in eV, the Planck mass ismPc21.6×10−19'1.2×1028eV.(24)The proton mass is about a GeV, or 109eV, so the Planck mass is about 19 orders ofmagnitude larger!1.4 (c)For two particles of massm1andm2and electric chargesq1andq2, the ratio of the electricforce~FEto the gravitational force~Fgbetween them is|~FE||~Fg|=14π0GN|q1||q2|m1m2,(25)where0is the permittivity of free space.The proton and electron both have an electriccharge magnitude of the fundamental unit of chargee= 1.6×10−19, so plugging in numbers,the ratio of forces is|~FE||~Fg| '14π(8.85×10−12) (6.67×10−11)(1.6×10−19)2(1.67×10−27) (9.11×10−31)'2.2×1039.(26)1.5Expansion of the Universe1.5 (a)The CMB has a temperature of 2.7 K. With Boltzmann’s constantkB, we can turn this intoa corresponding energy. We findECMB=kBT= 2.7×1.38×10−23J'3.7×10−23J.(27)To determine the energy in eV, we divide by 1.6×10−19:ECMB'3.7×10−231.6×10−19eV'2.3×10−4eV.(28)1.5 (b)The ground state energy of hydrogen isEg=−13.6 eV. So, when the temperature of theuniverse was less than the energy of 13.6 eV, electrons and protons could become bound andform hydrogen. To determine this temperature, we work backward from the steps of the firstpart of this problem, multiplying by the factor 1.6×10−19and then dividing bykB. We findTrecomb.= 1.6×10−19|Eg|kBK'1.6×105K.(29)9

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1.5 (c)We want to calculate the ratio of the wavelength of CMB photons observed today,λtoday, bythe wavelength at recombination,λrecomb.. This ratio isλtodayλrecomb.=frecomb.ftoday=Erecomb.Etoday=Trecomb.Ttoday.(30)In this chain of equalities, we used the fact that wavelengthλis inversely proportional tofrequencyfand the frequency of light is proportional to its energy. From what was developedin the previous parts, the energy of the photons is proportional to its temperature. So, theredshift factor is just the ratio of the temperature at recombination to the temperaturetoday:λtodayλrecomb.=Trecomb.Ttoday'1.6×1052.7'5.9×104.(31)As mentioned in the problem, this is a factor of about 30 larger than the true result whenthermodynamics are properly taken into account.1.6Decay Width of the Z boson1.6 (a)From the plot of Fig. 1.5, the maximum value of the peak of the distribution is about 32nb.Therefore, half of this is 16 nb.The lower point of the distribution at which it takesa 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, or2.5 GeV.1.6 (b)To determine the lifetime in seconds of theZboson from its width, we need to relate thewidth 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 1.6×10−19so that the lifetime in seconds is∆t'ℏ∆E'1.05×10−34(2.5×109)·(1.6×10−19)'2.6×10−25s.(33)We also needed to include a factor of 109to account for the fact that the width is 2.5 GeV =2.5×109eV.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 livesforever. Correspondingly, if the width gets very large, then the lifetime approaches 0: theparticle decays instantly.1.7Decay of Strange HadronsThe Ω−hadron travels about 3 cm = 0.03 m before decaying which corresponds to a lifetimeby multiplication by its velocity. While its velocity is not known, it will be an appreciablefraction of the speed of lightc, so we can just assume that it isc. Dividing byc, the lifetimeτisτ'0.033×108s'10−10s.(34)To convert to the decay width ∆E, we use the energy-time uncertainty relationship, and thefactor 1.6×10−19to convert to eV. We then find∆E=ℏτ'1.05×10−3410−10·(1.6×10−19) eV'6.6×10−5eV.(35)1.8PDG Review1.8 (a)From the PDG, the lower bound on the lifetime of the proton is 2.1×1029years. To havea reasonable probability to observe one proton decay in a year, you would need at least2.1×1029protons, if the lifetime were exactly at the lower bound. Water, H2O, consists of10 protons (and 8 neutrons), so we would need to observe 2.1×1028water molecules for ayear. The volume of this amount of water can be found by first identifying the number ofmoles of water:2.1×10286.02×1023mol'3.5×104mol.(36)Because water consists of 18 total protons and neutrons, this amount of water has a mass of(3.5×104)·18 g'630 kg.(37)Water has a density of 1000 kg/m3, so the total volume of water needed is about6301000 m3= 0.63 m3.(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 theW,Z, and Higgs bosonsand the top quark are:mW= 80.379 GeV,(39)mZ= 91.1876 GeV,(40)mH= 125.18 GeV,(41)mt= 173.0 GeV.(42)The mass of the proton or neutron is approximately 1 GeV (actually slightly less), and so thevalue of the mass in GeV can be approximately used to identify the atomic mass of elementswith about the same mass. For example, Krypton has an atomic mass of about 83, which isclose to the mass of theWboson. Zirconium has an atomic mass of about 91, close to themass of theZboson. Tellurium has an atomic mass of about 127, close to the mass of theHiggs boson. Finally, Ytterbium has an atomic mass of about 174, close to the mass of thetop quark.Students may find a slightly different selection of elements from a more precise accountingof 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 ofFig. 1.6. Again, we use the “Particle Listings” section of the PDG, and we have to do a bitof sleuthing to identify all the particles by their symbols. Their masses are:mK−= 493.677 MeV,(43)mΩ−= 1672.43 MeV,(44)mK0= 497.611 MeV,(45)mπ−= 139.57061 MeV,(46)mΞ0= 1314.82 MeV,(47)mK+= 493.677 MeV,(48)mΛ0= 1115.683 MeV,(49)mp= 938.2720813 MeV.(50)The width, or lifetime, from the PDG of the Ω−baryon is 0.821×10−10s, which is veryclose to our very simple estimate!1.9InSpire and arXiv1.9 (a)At InSpire, we can search for Noether’s papers and find her most highly-cited paper.Infact, InSpire only has one of her papers listed (all others are pure mathematics), which is12

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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-citedpapers 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. Nauk161, no. 5,61 (1991)].2Special Relativity2.1Properties of Lorentz TransformationsWe’re asked to verify that the matrixΛνμ=γ00γβ01000010γβ00γ(51)leaves the metric invariant:ΛᵀηΛ =η .(52)First, note that the matrix Λ is symmetric: Λᵀ= Λ. So, multiplying from the left, we haveΛᵀη=γ00γβ01000010γβ00γ10000−10000−10000−1=γ00−γβ0−10000−10γβ00−γ.(53)Continuing, multiplying by Λ on the right producesΛᵀηΛ =γ00−γβ0−10000−10γβ00−γγ00γβ01000010γβ00γ(54)=γ2(1−β2)0000−10000−10000−γ2(1−β2)=η .Recall that 1−β2=γ−2, from which the result follows.13

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2.2RapidityThe rapidityyis defined to bey= 12 logE+pzE−pz.(55)We can perform a Lorentz boost along the ˆzaxis by a velocityβby multiplication of themomentum four-vector by a Lorentz matrix:γ00γβ01000010γβ00γE00pz=γ(E+βpz)00γ(pz+βE).(56)Therefore, under a Lorentz transformation, the rapidity transforms asy→12 logγ(E+βpz) +γ(pz+βE)γ(E+βpz)−γ(pz+βE) = 12 log (1 +β)(E+pz)(1−β)(E−pz) =y+ 12 log 1 +β1−β .(57)That is, under a Lorentz boost along the ˆzaxis, the rapidity transforms additively.2.3Lorentz-Invariant MeasureUnder a Lorentz transformation, the coordinate four-vectorxμtransforms tox′μ= Λμνxν.(58)Then, under a Lorentz transformation, the coordinate measured4xtransforms asd4x′=|J|d4x ,(59)whereJis the Jacobian formed from the determinant of the derivative matrix:J= det∂x′μ∂xν.(60)From the Lorentz transformation above in terms of the matrix Λ, this partial derivative is∂x′μ∂xν= Λμν.(61)Therefore, the Jacobian is just the determinant of the Lorentz-transformation matrix:J= det Λ.(62)By the definition of Λ, it satisfiesΛᵀηΛ =η ,(63)14

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which enables us to calculate the determinant of Λ. First, note that transposition doesn’tchange the determinant:det Λ = det Λᵀ.(64)Also, the determinant of the metricηis just the product of its non-zero elements:detη=−1.(65)Then, taking the determinant of the Lorentz transformation equation, we havedet(ΛᵀηΛ) =−(det Λ)2=−1.(66)Therefore,|det Λ|= 1 =J, and so after Lorentz transformation the measured4xis un-changed.2.4Properties of Klein-Gordon EquationThe 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 periodTto the timetand demanding that thefieldφ(x) is unchanged:e−i(E(t+T)−~p·~x)=e−i(Et−~p·~x),(68)or that exp[−iET] = 1. Then, the periodT= 2π/Eand so the frequencyf=E/(2π). Asimilar procedure can be used to determine the wavelength of the solution. The wavelengthλis thenλ= 2π|~p|.(69)It then follows that the phase velocity is the just the product of the frequency and wavelength:v=λf=E|~p|.(70)2.5Maxwell’s EquationsGauss’s law has already been identified as the 0thcomponent of the equations of motion of∂μFμν=Jν.(71)Now, let’s takeν=i, a spatial coordinate.Then,Jiis theithcomponent of the currentvector and the left side of this equation is∂μFμi=−∂0F0i+∂iFii+∂jFji+∂kFki=−∂Ei∂t+(∂Bk∂j−∂Bj∂k).(72)15

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