Theoretical (Elementary) Particle Physics (Summer 2017)
ohl:Wed Jul 19 14:02:45 CEST 2017subject to change!Theoretical (Elementary) Particle Physics(Summer 2017)Thorsten OhlInstitut für Theoretische Physik und AstrophysikUniversität WürzburgAm Hubland97070 WürzburgGermanyPersonal Manuscript!Use at your own peril!July 19, 2017i
ohl:Wed Jul 19 14:02:45 CEST 2017Abstract Fundamentale Teilchen und Kräfte Symmetrien und Gruppen Quarkmodell Grundlagen der Quantenfeldtheorie Eichtheorien Spontane Symmetriebrechung Elektroschwaches Standardmodell Quantenchromodynamik Erweiterungen des Standardmodellssubject to change!i
ohl:Wed Jul 19 14:02:45 CEST 2017subject to change!iContents1 Introduction11.1 Literature . . . . . . . . . . . . . . . . . . . . . . . . . . .1Lecture 01: Tue, 25. 04. 20171.1.1 Elementary Particle Physics and Standard Model . .11.1.2 Quantum Field Theory . . . . . . . . . . . . . . . .11.1.3 Group Theory . . . . . . . . . . . . . . . . . . . . .21.2 The Setting . . . . . . . . . . . . . . . . . . . . . . . . . . .21.2.1 Dramatis Personae . . . . . . . . . . . . . . . . . .21.2.2 Place . . . . . . . . . . . . . . . . . . . . . . . . . .31.2.3 Tools . . . . . . . . . . . . . . . . . . . . . . . . . .41.2.4 Approaches . . . . . . . . . . . . . . . . . . . . . . .41.3 The Frontier (as of today): LHC . . . . . . . . . . . . . . .52 Fundamental Particles and Forces2.1 What Is an Elementary Particle? . .2.1.1 Quantum Numbers . . . . . .2.2 Fundamental Interactions . . . . . .2.2.1 Gravity . . . . . . . . . . . .2.2.2 Electromagnetism . . . . . .2.2.3 The Strong Force . . . . . . .2.2.4 Weak Interactions . . . . . .3 Symmetries and3.1 Symmetries .3.2 Lie Groups .3.3 Lie Algebras3.4.66777778Groups9. . . . . . . . . . . . . . . . . . . . . . . . . .9. . . . . . . . . . . . . . . . . . . . . . . . . . 10. . . . . . . . . . . . . . . . . . . . . . . . . . 11Lecture 02: Wed, 26. 04. 2017Representations . . . . . . . . . . . . . . . . . . . . . . . . 123.4.1 Irreducible Representations . . . . . . . . . . . . . . 143.4.2 Direct Sums . . . . . . . . . . . . . . . . . . . . . . 143.4.3 Tensor Products . . . . . . . . . . . . . . . . . . . . 15
ohl:Wed Jul 19 14:02:45 CEST 2017subject to change!ii3.4.43.5Complex Conjugation . . . . . . . . . . . . . . . . . 16Lecture 03: Tue, 02. 05. 2017Lorentz and Poincaré Group . . . . . . . . . . . . . . . . . . 173.5.1 Lorentz Group . . . . . . . . . . . . . . . . . . . . . 18Lecture 04: Wed, 03. 05. 20173.5.2 Poincaré Group . . . . . . . . . . . . . . . . . . . . 243.5.3 Extensions of the Poincaré Group . . . . . . . . . . . 254 Quark Model4.1 The Particle Zoo . . . . . . .4.2 Isospin . . . . . . . . . . . .4.2.1 Strong Interactions vs.4.2.2 Dublets and Triplets .4.34.427. . . . . . . . . . . . . . . . . 27. . . . . . . . . . . . . . . . . 27Electromagnetism . . . . . . . 27. . . . . . . . . . . . . . . . . 28Lecture 05: Tue, 09. 05. 2017Lecture 06: Wed, 10. 05. 2017Eight-Fold Way . . . . . . . . . . . . . . . . . . . . . . . . 354.3.1 SU(3) . . . . . . . . . . . . . . . . . . . . . . . . . . 364.3.2 Structure Constants . . . . . . . . . . . . . . . . . . 374.3.3 Representations . . . . . . . . . . . . . . . . . . . . 384.3.4 Cartan Subalgebra and Rank . . . . . . . . . . . . . . 384.3.5 Roots and Weights . . . . . . . . . . . . . . . . . . . 39Lecture 07: Tue, 16. 05. 20174.3.6 Back to SU(3) . . . . . . . . . . . . . . . . . . . . . 414.3.7 Quarks, Octets and Decuplets . . . . . . . . . . . . . 43Lecture 08: Wed, 17. 05. 20174.3.8 Tensor Methods . . . . . . . . . . . . . . . . . . . . 494.3.9 Gell-Mann–Okubo Formula . . . . . . . . . . . . . . 51Lecture 09: Tue, 23. 05. 2017Heavy Quarks . . . . . . . . . . . . . . . . . . . . . . . . . 554.4.1 Quarkonia . . . . . . . . . . . . . . . . . . . . . . . 555 Gauge Theories5.1 Basics of Quantum Field Theory5.1.1 Classical Field Theory . .5.1.2 Quantum Field Theory .5.1.3 Pathintegral and Feynman56. . . . . . . . . . . . . . . 56. . . . . . . . . . . . . . . 56. . . . . . . . . . . . . . . 58Rules . . . . . . . . . . . 59Lecture 10: Wed, 24. 05. 2017
ohl:5.25.35.4subject to change!Wed Jul 19 14:02:45 CEST 2017Gauge5.2.15.2.25.2.3Invariant Actions . . . .Global TransformationsNoether’s Theorem . . .Local Transformations .5.2.4 Covariant Derivative . .5.2.5 Field Strength . . . . .5.2.6 Building Blocks . . . .Quantization . . . . . . . . . .5.3.1 Perturbative Expansion5.3.2 Propagator . . . . . . .5.3.3 Faddeev-Popov Procedure . .5.3.4 Feynman Rules . . . . . . . .Massive Gauge Bosons . . . . . . . .iii. . . . . . . . . . . . 61. . . . . . . . . . . . 61. . . . . . . . . . . . 63. . . . . . . . . . . . 67Lecture 13: Tue, 30. 05. 2017. . . . . . . . . . . . . 67. . . . . . . . . . . . . 69. . . . . . . . . . . . . 70. . . . . . . . . . . . . 70. . . . . . . . . . . . . 70. . . . . . . . . . . . . 71Lecture 14: Wed, 31. 05. 2017. . . . . . . . . . . . . 73. . . . . . . . . . . . . 75. . . . . . . . . . . . . 76Lecture 15: Tue, 05. 06. 20176 Spontaneous Symmetry Breaking806.1 Goldstone’s Theorem . . . . . . . . . . . . . . . . . . . . . . 806.2 Higgs Mechanism . . . . . . . . . . . . . . . . . . . . . . . 83Lecture 16: Tue, 13. 06. 20176.2.1 Unitarity Gauge . . . . . . . . . . . . . . . . . . . . 836.2.2 Rξ -Gauge . . . . . . . . . . . . . . . . . . . . . . . . 857 (Elektroweak) Standard Model887.1 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . 887.2 Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89Lecture 17: Wed, 14. 06. 20177.2.1 Solution . . . . . . . . . . . . . . . . . . . . . . . . 907.3 SU(2)L U(1)Y /U(1)Q . . . . . . . . . . . . . . . . . . . . 907.3.1 Matter Fields . . . . . . . . . . . . . . . . . . . . . . 917.3.2 Higgs Fields . . . . . . . . . . . . . . . . . . . . . . 93Lecture 18: Tue, 20. 06. 20177.3.3 Yukawa Couplings . . . . . . . . . . . . . . . . . . . 98Lecture 19: Wed, 21. 06. 20177.3.4 Interactions . . . . . . . . . . . . . . . . . . . . . . 103Lecture 20: Tue, 27. 06. 20177.4 GIM-Mechanism . . . . . . . . . . . . . . . . . . . . . . . . 1077.5 K 0 -K 0 Mixing, etc. . . . . . . . . . . . . . . . . . . . . . . 109Lecture 21: Wed, 28. 06. 2017
ohl:7.67.7subject to change!Wed Jul 19 14:02:45 CEST 20177.5.17.5.27.5.37.5.47.5.5Box diagrams . . . . . . .Mixing . . . . . . . . . . .Experimental ObservationsCP Conservation . . . . .K 0 K 0 Oscillations . . .7.5.6CP Violation. . . . . . . .7.5.7 B 0 -B 0 and D0 -D0 . . . . . .Higgs Production and Decay . . . . .Anomaly Cancellation . . . . . . . .iv. . . . . . . . . . . . . 109. . . . . . . . . . . . . 110. . . . . . . . . . . . . 111. . . . . . . . . . . . . 112. . . . . . . . . . . . . 113Lecture 21: Wed, 24. 06. 2015. . . . . . . . . . . . . 114Lecture 22: Tue, 04. 07. 2017. . . . . . . . . . . . . 116. . . . . . . . . . . . . 117. . . . . . . . . . . . . 119Lecture 23: Wed, 05. 07. 20178 Quantum Chromo Dynamics1218.1 Lowest Order Perturbation Theory . . . . . . . . . . . . . . 1218.2 Lattice Gauge Theory . . . . . . . . . . . . . . . . . . . . . 1218.3 Asymptotic Freedom . . . . . . . . . . . . . . . . . . . . . . 122Lecture 24: Tue, 11. 07. 20178.3.1 Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . 122Lecture 25: Wed, 12. 07. 20178.3.2 Parton Model . . . . . . . . . . . . . . . . . . . . . 1248.4 Hadronization . . . . . . . . . . . . . . . . . . . . . . . . . 1249 Extensions of the Standard Model9.1 Gravity . . . . . . . . . . . . . . . .9.2 Naturalness . . . . . . . . . . . . . .9.2.1 Proposed Solutions . . . . . .9.3Dark Matter . . .9.3.1 Candidates9.4 Grand Unification9.5 New Particles . .125Lecture 26: Tue, 18. 07. 2017. . . . . . . . . . . . . 125. . . . . . . . . . . . . 125. . . . . . . . . . . . . 126Lecture 27: Wed, 19. 07. 2017. . . . . . . . . . . . . 126. . . . . . . . . . . . . 127. . . . . . . . . . . . . 127. . . . . . . . . . . . . 128A Acronyms129B Bibliography131
ohl:Wed Jul 19 14:02:45 CEST 2017subject to change!vVorbemerkungDieses Manuskript ist mein persönliches Vorlesungsmanuskript, an vielenStellen nicht ausformuliert und kann jede Menge Fehler enthalten. Es handeltsich hoffentlich um weniger Denk- als Tippfehler, trotzdem kann ich deshalbich keine Verantwortung für Fehler übernehmen. Zeittranslationsinvarianzist natürlich auch nicht gegeben . . .Dennoch, oder gerade deshalb, bin ich für alle Korrekturen und Vorschlägedankbar!OrganisatorischesKontakt Büro: 22.02.009 (Hubland Nord, Emil-Hilb-Weg 22, 2. Stock) Sprechstunde: nach Vereinbarung Mail: ohl@physik.uni-wuerzburg.de URL: http://physik.uni-wuerzburg.de/ohl/Aktuelle InformationenVorlesungs-URL zu lang, einfach auf http://physik.uni-wuerzburg.de/ohl/ gehen und den Links folgen,dort auch Inhaltsangabe.Übungsgruppen #01: Montag, 10:00 UhrÜbungszettel .
ohl:Wed Jul 19 14:02:45 CEST 2017subject to change!1—1—Introduction1.1LiteratureLecture 01: Tue, 25. 04. 20171.1.1Elementary Particle Physics and Standard ModelAdvanced Howard Georgi: Weak Interactions and Modern Particle Theory, Dover,2009. NB: the author makes a PDF file of an updated version availableat his home page: http://www.people.fas.harvard.edu/ hgeorgi/weak.pdf John F. Donoghue, Eugene Golowich, Barry R. Holstein: Dynamics ofthe Standard Model, Cambridge University Press, 1992.1.1.2Quantum Field TheoryIntroductory Micheal E. Peskin, Daniel V. Schroeder: An Introduction to QuantumField Theory, Addison-Wesley Publishing Company, 1995. Claude Itzykson, Jean-Bernard Zuber: Quantum Field Theory, McGrawHill, 1990.Advanced Steven Weinberg: The Quantum Theory of Fields. Volume I: Foundations, Cambridge University Press, 1995.
ohl:Wed Jul 19 14:02:45 CEST 2017subject to change!2 Steven Weinberg: The Quantum Theory of Fields. Volume II: ModernApplications, Cambridge University Press, 1996.1.1.3Group TheoryIntroductory Howard Georgi: Lie Algebras in Particle Physics, 2nd ed., PerseusBooks, 1999.Unorthodox Predrag Cvitanović: Group Theory: Birdtracks, Lie’s, and ExceptionalGroups, Princeton University Press, 2008. NB: the author makes aPDF file of the book available at http://birdtracks.eu/1.21.2.1The SettingDramatis PersonaeStable ParticlesThese have never been observed to decay if left alone electrons (e ) and positrons (e ) photons (γ) protons (p) and anti-protons (p̄)and γ, e and p, together with neutrons, make up all “normal matter”.Almost Stable ParticlesThese live long enough to leave macroscopic O(1 m) tracks in detectors: neutrons (n) and anti-neutrons (n̄) muons (µ ) and antimuons (µ )
ohl:subject to change!Wed Jul 19 14:02:45 CEST 20173Unstable Particles (a. k. a. Resonances)Everything else decays too quickly to be seen as a track in detectors.The exchange of a particle with mass M corresponds to an amplitudeµ e ip2 M 2 i (1.1)µ e and a cross sectioniσ(s) s M22 1(s M 2 )2(1.2)with an unphysical singularity at s M 2 . A more careful computationreveals a finite width withii p2 M 2 i p2 M 2 iM Γ(1.3)and a Breit-Wigner resonance shapeσ(s) 2is M2 iM Γ 1(s M 2 )2 M 2 Γ2.(1.4)The mass M of the particle can then be measured as the location of thepeak of the cross section and the lifetime τ of the particle as the inverse ofthe width Γ 1/τ of the resonance. A typical example is the spectrum ofresonances decaying into muon pairs at LHC, shown in figure 1.1. Obviously,these resonances must correspond to uncharged particles and we can expectmore resonances in other channels, corresponding to charged particles.1.2.2PlaceTypical energies in nuclear reactions are O(10 MeV) corresponding to lengthscales O(10 fm) O(10 14 m), using the conversion factor c 197 MeV fm .(1.5)“Interesting” elementary particle starts with O(1 GeV) and we are now testing the “terascale” O(1 TeV) for the first time.
Wed Jul 19 14:02:45 CEST 2017106J/,subject to change!Events / ( 0.1 GeV/c2 )Events / GeVohl:'4CMS, s 7 TeV45000Lint 40 pb-140000 2.435000 100 1010.511 -11.512mass (GeV/c2)Z1031028.5CMSs 7 TeVLint 40 pb -1110102Dimuon mass (GeV/c2 )Figure 1.1: Resonances in pp µ µ X measured by the CMS experimentat LHC in 2010 [1].1.2.3ToolsExperimentAccelerators and colliders natural (cosmic) man made (LHC etc.)Theory Quantum Field Theory (QFT) for computing cross sections and decayrates group theory for organizing particles and interactions1.2.4ApproachesThere are two complementary approaches that are both required for progressin our understanding of the microcosmos
ohl:Wed Jul 19 14:02:45 CEST 2017subject to change!5Bottom-Up1. write down the most general mathematically consistent interaction ofthe observed particles, consistent with the observed symmetries andconservation laws (cf. Noether theorem)2. fit the free parameters to observations3. compute cross section and decay rates for the observed particles4. compare with experiment5. if necessary, add new particles and repeatTop-Down1. propose an improved microscopic model of elementary particles andtheir interactions2. compute cross section and decay rates for the observed particles, whichmight be bound states of the elementary particles3. compare with experiment4. repeat1.3The Frontier (as of today): LHCIn figure 1.1, the Standard Model (SM) predictions are tested by a singleexperiment over more than two orders of magnitude in energy and five ordersof magnitude in cross section.As we are speaking, the LHC is restarting for “Run 2”, which will probethe SM predictions well into the terascale.
ohl:Wed Jul 19 14:02:45 CEST 2017subject to change!6—2—Fundamental Particles and Forces2.1What Is an Elementary Particle?The answer to this natural question is both trivial and subtle:a particle is considered elementary, if and only if (iff) there is noevidence that it is composite, i. e. there no finite spacial extend it can not be broken apart .Therefore, the category of elementary particles is not constant in time a little more than a century ago (before Rutherford), atoms whereconsidered elementary between 1910 and 1950, electrons, photons, protons and neutrons wereconsidered elementary in the early 1950s, Robert Hofstadter discovered by elastic electronscattering of protons (hydrogen) that protons have a size of roughly1 fm 10 15 m as of today, electrons and photons still qualify as elementaryStill, the term elementary particle physics also refers to particles that we nowknow to be unstable or composite, such as protons, neutrons, other baryonsand mesons.
ohl:Wed Jul 19 14:02:45 CEST 20172.1.1subject to change!7Quantum NumbersAccording to our observations, elementary particles are completely indistinguishable: if the have the same quantum numbers, their states must be eithersymmetric (bosons) or antisymmetric (fermions) under permutations.Therefore an elementary particle is completely characterized by its representation of the Poincaré group, i. e. mass, spin and parities underspace, time and charge inversion electric charge other more exotic charges: isospin, color, etc.Of these, the mass, spin and the parities correspond to spacetime symmetries,while the rest are called internal symmetries.In the absence of gravity, i. e. in a flat space time, elementary particles canonly have the spins 0 (scalar), 1/2 (spinor) and 1 (vector), while compositeparticles can have any spin (cf. Clebsh–Gordan decomposition). If we includegravity, particles with spin 3/2 and 2 (gravitons) become possible.2.2Fundamental InteractionsAs of today all interactions among elementary particles are described by fourfundamental interactions:2.2.1GravityThis is the only interactions felt by all elementary particles, since it affectsspace-time itself. Unfortunately, we don’t have a good quantum mechanicaldescription yet. Fortunately, its effects on elementary particles are so weakat accessible energy scales that it can savely be ignored.2.2.2ElectromagnetismThis is described by Quantum Electro Dynamics (QED) to an incredibleprecision and matches to electromagnetism in the classical limit.2.2.3The Strong ForceThis has no classical analog and affects only baryons and mesons.
ohl:Wed Jul 19 14:02:45 CEST 20172.2.4subject to change!Weak InteractionsThis also has no classical analog.8
ohl:Wed Jul 19 14:02:45 CEST 2017subject to change!9—3—Symmetries and Groups3.1SymmetriesIf a charge Q commutes with the Hamiltonian H[H, Q] 0 ,(3.1)it is conserveddQ i[H, Q] 0 .(3.2)dtIn addition, if such a charge relates two eigenstates of the Hamiltonian HQ 1i 2i(3.3)H ni En ni ,(3.4)withthenE2 2i H 2i HQ 1i QH 1i QE1 1i E1 Q 1i E1 2i(3.5)i. e.E1 E2(3.6)and the states 1i and 2i are degenerate.Therefore we will have multiplets of degenerate states { ii}i I Z , whenever these states form a representation (section 3.4) of a Lie algebra (section 3.3) of conserved charges {Qj }j J Z[H, Qi ] 0X[Qi , Qj ] ifijk Qk ,k J(3.7a)(3.7b)
ohl:subject to change!Wed Jul 19 14:02:45 CEST 201710i. e.Qi ji X[r(Qi )]jk ki .(3.8)k JSince2H 2 M 2 P ,(3.9)the above reasoning translates from energy levels to masses.3.2Lie GroupsIn physics1 , symmetries are described as Groups (G, ) with G a set and an inner operation :G G G(3.10)(x, y) 7 x ywith1. closure: x, y G : x y G,2. associativity: x (y z) (x y) z,3. identity element: e G : x G : e x x e x,4. inverse elements: x G : x 1 G : x x 1 x 1 x e .Many examples in physics permutations reflections parity translations rotations Lorentz boosts Runge–Lenz vector isospin .1And mathematics!
ohl:Wed Jul 19 14:02:45 CEST 2017subject to change!11Particularly interesting are Lie Groups, i. e. groups, where the set isa differentiable Manifold and the composition is differentiable w. r. t. bothoperands.Note that the choice of coordinates is not relevant:() 0 ηcosh η sinh ηB b1 (η) exp η R η 0 sinh η cosh η)( 11 ββ ] 1, 1[(3.11) b2 (β) p1 β 2 β 1Both times we have the set of all real symmetric 2 2 matrices with unitdeterminant. The composition laws are given by matrix multiplication2 :b1 (η) b1 (η 0 ) b1 (η)b1 (η 0 ) b1 (η η 0 ) β β000b2 (β) b2 (β ) b2 (β)b2 (β ) b2.1 ββ 03.3(3.12a)(3.12b)Lie AlgebrasLecture 02: Wed, 26. 04. 2017A Lie algebra (A, [·, ·]) is a K-vector space3 with a non-associative antisymmetric bilinear inner operation [·, ·]:[·, ·] : A A A(a, b) 7 [a, b]with1. closure: a, b A : [a, b] A,2. antisymmetry: [a, b] [b, a]3. bilinearity: α, β K : [αa βb, c] α[a, c] β[b, c]4. Jacobi identity: [a, [b, c]] [b, [c, a]] [c, [a, b]] 02NB: β 1 β 0 1 3K R or Cβ β0 11 ββ 0(3.13)
ohl:Wed Jul 19 14:02:45 CEST 2017subject to change!Since A is a vector space, we can choose a basis and writeX[ai , aj ] Cijk ak .12(3.14)kA Lie algebra is called simple, if it has no ideals besides itself and {0}.Remarkably, all simple Lie algebras are known:so(N ), su(N ), sp(2N ), g2 , f4 , e6 , e7 , e8(3.15)with N N.The infinitesimal generators of a Lie group form a Lie algebra. Vice versa,the elements of a Lie algebra can be exponentiated to obtain a Lie group (notnecessarily the same, but a cover of the original group).3.4RepresentationsA group homomorphism f is a mapf : G G0x 7 f (x)(3.16)between two groups (G, ) and (G0 , 0 ) that is compatible with the groupstructuref (x) 0 f (y) f (x y)(3.17)and thereforef (e) e0f (x 1 ) (f (x)) 1 .(3.18a)(3.18b)A Lie algebra homomorphism φ is a mapφ : A A0a 7 φ(a)(3.19)between two Lie algebras (A, [·, ·]) and (A0 , [·, ·]0 ) that is compatible with theLie algebra structure[φ(a), φ(b)]0 φ([a, b]) .(3.20)NB: these need not be isomorphisms: f (x) e0 , x is a trivial, but welldefined group homomorphism and φ(a) 0, a is a similarly trivial but als
1.1.2 Quantum Field Theory Introductory Micheal E. Peskin, Daniel V. Schroeder: An Introduction to Quantum Field Theory, Addison-Wesley Publishing Company, 1995. Claude Itzykson, Jean-Bernard Zuber: Quantum Field Theory, McGraw-Hill, 1990. Advanced Steven Weinberg: The Quantum Theory of Fields. Volume I: Founda-tions, Cambridge University Press .
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Usually z 1 in elementary particle physics, but not in nuclear physics, heavy ion physics or cosmic rays. methods of particle identification : Measure the bending radius ρin a magnetic field B (p B ): mv2 ρ z·e·v·B ρ p zeB γm0βc z with p: momentum; β v c; γ q 1 1 v2 c2. Physics of Particle Detection .
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INTRODUCTION TO PARTICLE PHYSICS PHYS 5380 Recommended reading: Elementary level D.H. Perkins, Introduction to High Energy Physics Medium Level David Griffith, Introduction to Elementary Particles Advanced F. Halzen and A. Martin, Quarks and Leptons Gordon Kane, Modern Elementary Particle Physics, updated edition Experimental techniques
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