PHYS 6610: Graduate Nuclear and Particle Physics I H. W. Grießhammer INS Institute for Nuclear Studies The George Washington University Institute for Nuclear Studies Spring 2018 II. Phenomena 5. e + e − → Leptons and Quarks Or: Why We Believe in Things We Don’t See References: [PRSZR 9.1/3; PRSZR 15/16 (cursorily); HG 10.9, 15.1-7; HM 11.1-3; Tho 9.6] PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University II.5.0
(a) Recap e + e − → µ + µ − : Massless Point-Fermions � = ( Z α ) 2 d σ � ( 1 + cos 2 θ ) � d Ω � cm 4 s Ang. distrib. characteristic of spin- 1 2 . s = q 2 > 0 : timelike γ Z k ′ տ ր p ′ Final state not electrons, ↑ q has quantum numbers of virtual photon: e − ( k ) ր տ e + ( p ) I ( J PC ) = 0 or 1 ( 1 −− ) [Tho 6.7] 137 , simple to interpret, e + e − collider cheap 1 � � � + + + � � � − − − Directly probes only charges, not strong int. σ cm = 4 π ( Z α ) 2 21 . 7nb = Z 2 E e cm [ GeV ] 3 s For massless point-fermions X of charge Z X : R : = σ ( e + e − → X ¯ X ) Z 2 ∑ → X 4 πα 2 / ( 3 s ) finals X with √ s ≥ 2 M X PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University II.5.1
(b) Leptoproduction and Lepton Universality Threshold √ s min = 2 M l : muon M µ = 0 . 106 GeV (1936), tau lepton M τ = 1 . 777 GeV (1975) Lifetime τ ( τ − → e − ¯ ν e ν τ or µ − ¯ ν µ ν τ ) = 3 × 10 − 13 s ≫ τ elmag , strong = ⇒ weak decay Even at γ = E ≈ 100GeV ≈ 50 , τ lepton travels c γτ τ = 10 − 2 m before decay = ⇒ not in detector M τ Experiments at E ≫ M τ , M µ : R ( µ + µ − or τ + τ − ) = 1 = ⇒ Z µ = Z τ = 1 = Z e [PRSZR] = ⇒ Lepton Universality Hypothesis: Leptons couple with same form & strengths, and differ only by mass & charge (thresholds etc. different, but not fundamental couplings). PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University II.5.2
Is Lepton Universality Broken? Update 2018. BaBar at SLAC 2012: branching ratio B → µ or τ [Phys. Rev. Lett. 109 (2012) 101802] LHCb at CERN 2015: branching ratio D , D ∗ → µ or τ [Phys. Rev. Lett. 115 (2015) 111803] [Heavy Flavour Averaging Group 2018] Lepton Universality may be broken: Small ( 10 − 2 ) but significant ( 3 . 9 σ ?) and important (Baryogenesis). = ⇒ Beyond-Standard-Model Physics? PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University II.5.3
(c) e + e − → Hadrons : Overview non-resonant: well-reproduced by σ ∝ 1 s resonances: have quantum numbers of γ ∗ : I = 0 or 1 , J PC = 1 −− wide at low s , narrow at high s R = σ ( e + e − → hadrons ) σ ( e + e − → µ + µ − ) increases after each resonance region = ⇒ Hadron contains point-like, charged fermions with “small” masses. [PDG 2012 46.6] PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University II.5.4
(d) Nonresonant Hadron Production (High Energies) � = ( Z α ) 2 2 particle: d σ � ( 1 + cos 2 θ ) Produce point spin- 1 � d Ω � cm 4 s √ s = 34 GeV [Per 5.5] Angular distribution of 2-jet event consistent with 1 + cos 2 θ = ⇒ Evidence for point-fermions. PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University II.5.5
R Counts Quark Charges AND Colours R ( s ) : = σ ( e + e − → hadrons ) Z 2 ∑ σ ( e + e − → µ + µ − ) = i quarks i with 2 M i ≤ √ s = 10 9 � �� � � 2 � 2 6 9 + � 1 � 2 10 bottom = 11 9 + 3 charm 3 9 = 6 R = N c ∑ Z 2 q = 3 ∑ Z 2 q universal hidden property 9 � �� � q q � 2 � 2 � 2 � 2 � 1 � 1 = ⇒ each quark flavour in 3 variants: colours up + down + 3 3 3 strange PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University II.5.6
Hadronisation from Vacuum: Avoiding Free Quarks & Gluons q pair carries energy E ≫ m q ; fly in opposite directions . No free quarks seen. Each quark of initial q ¯ = ⇒ Generate light q ¯ q pairs out of vacuum. Rearrange to dress bare quarks into baryons & mesons : timescale ≫ pair-production = ⇒ Production & hadronisation: 2-step process: incoherent sum of q ¯ q -pair productions d σ d Ω ( ee → hX ) d σ = ∑ q ) [ D h q ( z )+ D h d Ω ( ee → q ¯ q ( z )] ¯ q which are weighted by Quark-fragmentation functions q ( z = E h D h q , D h ) related to PDFs q ( x ) ¯ E q by crossing & time-reversal symmetries. PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University II.5.7
(e) Resonant Hadron Production at Low Energies R Broad J PC = 1 −− resonances, τ ∼ 10 − [ 22 ... 24 ] s = ⇒ strong process. √ s [ GeV ] [PDG 2012 46.7] ω 0 ( 782 MeV ) : decay → π + π 0 π − ; no isospin partners = ⇒ I = 0 ρ 0 ( 770 MeV ) : decay → π + π − ; isospin partners ρ ± , 0 = ⇒ I = 1 spin-isospin-quark content e.g. | ρ + � = −| u ↑ ¯ d ↑ � Resonances in close proximity = ⇒ strong interference! = ⇒ Vector Meson Dominance (VMD) Model [ Sakurai 1960/69 ] : ρ , ω , φ N Elmag. dominated by these mesons, e.g. in γ N [HG 10.15] PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University II.5.8
Post-Dicting Vector Mesons (VMD Endorsement) Something interesting should indeed happen around 800MeV : Lowest hadron production threshold: √ s = 2 m π from e + e − → ππ . πγ coupling from pion form factor for space-like q 2 < 0 (see II.2.g): pion FF � �� � a 2 6 ց q π = − i e ( p ′ µ − k ′ µ ) a 2 − q 2 with a 2 = J µ π � ≈ ( 740MeV ) 2 (exp) F π ( q 2 < 0 ) � r 2 Apply crossing symmetry/analytic continuation into time-like region q 2 = s > 0 : ⇒ Expect pole/very large amplitude/resonance in J PC = 1 −− processes around = → q q 2 = s = a 2 ≈ ( 740MeV ) 2 F π ( q 2 > 0 ) Agrees with ω / ρ -meson quantum numbers and m ω ≈ m ρ ≈ 775MeV . But be careful: Exp. only gives rough form factor with uncertainties. = ⇒ Analytic continuation needs “reasonable” assumptions. PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University II.5.9
(f) φ Resonance: The Strange Quark φ ( 1019 MeV ) 85% → K + K − or K 0 ¯ K 0 − R √ s [ GeV ] [PDG 2012 46.7] Narrow resonance: Γ φ = 4 . 4 MeV since 2 m K = 990 MeV = ⇒ only 30 MeV of phase space! φ → πππ decay very small, although m φ − 3 m π = 600 MeV much bigger. = ⇒ Attribute to new quark flavour: Strange Quark ; strangeness S conserved in strong int. New charge formula: Q = Baryon + I 3 + S K + K − , K 0 ¯ K 0 isospin doublets = ⇒ 2 2 Gell-Mann–Nishijima relation Q s = − 1 , B s = 1 But strangeness of strange is S ( s ) = − 1 : That’s strange! (but a definition. . . ) 3 3 PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University II.5.10
Gell-Mann 1962 (g) Extending Isospin: The Eightfold Way good explanation: [Tho 9.6] “Tamed” the Particle Zoo in the 1960’s. Multiplet classification scheme still used for nomenclature. Interpret Strangeness (or Strong Hypercharge Y = S + B ) as quantum number, orthogonal to Isospin. � u � One Can Show: symmetry group in Nature extends from SU I ( 2 ) → SU flavour ( 3 ) acting on d : s = ⇒ Fundamental quark-triplet & anti-triplet. Ladder Operators I ± , U ± , V ± raise/lower along triangle sides. PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University II.5.11
Constructing Multiplets from The Fundamental Representations Combine Weight Diagrams like in SU ( 2 ) : Example: q ¯ q combinations give Meson Octet & Singlet. PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University II.5.12
Lowest-Mass Meson Octets: Natural Isospin Doublets K + K 0 & K − ¯ K 0 Constituent picture = ⇒ Gell-Mann–Okubo mass formula: m meson = M meson + ∑ m qi ? bind i m s ≈ 360MeV ? diff. ± 350MeV avg. 320MeV ≈ 1! ; in Excited Octet: ± 80MeV 850MeV ≈ 1 SU f ( 3 ) -breaking in Ground-State Octet: 10 Example Octet-Breaking in φ : m φ − 3 m π ≈ 600MeV ≫ m φ − 2 m K ≈ 30MeV but π decay tiny, m s = m K − m ρ while decays to 85% into Kaons ≈ 120MeV ? = ⇒ | φ � ≈ | s ¯ s � ] or ( m φ − m ρ ) / 2 1 u � + | d ¯ ≈ 120MeV ? and not | φ � = 3 [ | u ¯ d � + | s ¯ s � √ PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University II.5.13
π N / KN → X : Lowest-Mass Baryon Multiplets: Octet & Decouplet GMO: M baryon = M baryon + ∑ m qi bind i ⇓ linear increase m s ≈ 210MeV ? SU f ( 3 ) Breaking in Baryon Octet: ± 180MeV in Baryon Decouplet: ± 150MeV 1100MeV ≈ 1 1400MeV ≈ 1 6 10 ⇓ linear increase m s ≈ 140MeV ? Gell-Mann 1962: predict quantum numbers & mass of Ω − . Dedicated experiment found it. PHYS 6610: Graduate Nuclear and Particle Physics I, Spring 2018 H. W. Grießhammer, INS, George Washington University II.5.14
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