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Weak Interactioninteraction WeakWeak InteractionInteraction OutlineOutline Introduction to weak interactions Charged current (CC) interactions Neutral current (NC) Weak vector bosons W± and Z0 Weak charged interactions Beta decay Flavour changing charged current W± boson propagator Fermi coupling constant Parity violation Muon decay Decay rate /lifetime Lepton Universality W± boson couplings for leptons Tau decays Weak quark decays W± boson couplings for quarks Cabibbo angle, CKM mechanism Spectator model Nuclear and Particle Physics Franz Muheim 1 IntroductionIntroduction Weak Interactions Account for large variety of physical processes Muon and Tau decays, Neutrino interactions Decays of lightest mesons and baryons Z0 and W± boson production at √s ~ O(100 GeV) Natural radioactivity, fission, fusion (sun) Major Characteristics Long lifetimes Small cross sections (not always) “Quantum Flavour Dynamics” Charged Current (CC) Neutral Current (NC) mediated by exchange of W± boson Z0 boson Intermediate vector bosons MW = 80.4 GeV ± 0 W and Z have mass MZ = 91.2 GeV Self Interactions of W± and Z0 also W± and γ Nuclear and Particle Physics Franz Muheim 2 BetaBeta DecayDecay Weak Nuclear Decays See also Nuclear Physics Recall β+ = e+ β- = e- Continuous energy spectrum of e- or e+ Î 3-body decay, Pauli postulates neutrino, 1930 Interpretation Fermi, 1932 Bound n or p decay − n → pe ν e ⎛ − t ⎞ N(t) = N(0)exp⎜ ⎟ ⎝ τ n ⎠ p → ne +ν (bound p) τ n = 885.7 ± 0.8 s e 32 τ 1 = τ n ln 2 = 613.9 ± 0.6 s τ p > 10 y (p stable) 2 Modern quark level picture Weak charged current mediated by exchange of virtual W± boson Nuclear and Particle Physics Franz Muheim 3 WeakWeak ChargedCharged InteractionsInteractions Fundamental Vertex Weak Charged Current QED Flavour Changing Weak charged current changes lepton and quark flavours ± - W boson transforms νe into e , u into d-quark Only weak interaction is flavour changing Weak Charged Current ± W boson couples to weak charge gW of neutrino/electron or up/down quark pair Coupling strength ∝ gW Î For each vertex add factor gW to matrix element Probability for weak vertex 2 cross section or decay rate ∝ gW W± Boson Propagator W boson has mass MW Recall photons and gluons are massless 2 2 Î Add propagator term 1/(q –MW ) to matrix element/amplitude Nuclear and Particle Physics Franz Muheim 4 FermiFermi TheoryTheory ofof WeakWeak InteractionInteraction W± Boson Propagator 2 2 2 for small q << MW (q → 0) 2 2 2 1/(q –MW ) → 1/MW Propagator is constant Fermi Coupling Constant GF Weak coupling constant includes propagator 2 2 gW GF gW GF ∝ 2 exact = 2 Fermi MW 2 8MW different from QED & QCD Perkins, p 151&210 -2 gW dimensionless → units of GF are GeV 3 -5 -2 GF/(ħc) = 1.16637(1) ·10 GeV Range of Weak Interaction Massive exchange boson ↔ short range hc −18 Rweak = 2 ≈ 2⋅10 m = 0.002 fm << 0.1 fm MW c Analogous to Yukawa interaction Strength of Weak Interaction g 2 1 1 G & M ⇒ g = 0.66 ⇒ α = W = > α = F W W W 4π 29 em 137 Not intrinsically weak 2 at low q weak due to large MW αS ≈ 0.2 > αW ≈ 0.03 > αem ≈ 0.008 all running Will these meet at high energy? – unification? Nuclear and Particle Physics Franz Muheim 5 ParityParity ViolationViolation Parity P Intrinsic quantum number of particles Conserved in strong and electromagnetic interactions Kaon Decays Observe K+ → π+ π0 and K+ → π+ π- π+ decays Parity of K+ and π+ P(K + ) = P(π + ) = −1 + 0 L=0 Parity of final states P()π π = Pπ Pπ ()− 1 = 1 + − + L=0 Î Puzzle P()π π π = Pπ Pπ Pπ ()− 1 = −1 Parity Violation in Weak Interactions 1956 proposed by Lee and Yang 1957 experimentally confirmed by Wu et al 60 60 60 * − Measure e- from Co beta decay Co→ Ni e νe Polarised 60Co spin J aligned to magnetic field B at 0.01 K B,J B,J If parity conserved expect equal e- rates parallel and antiparallel to B-field and spin Observe asymmetry Î Parity violated Nuclear and Particle Physics Franz Muheim 6 MuonMuon DecayDecay Muon Fundamental lepton with mass mµ = 105.7 MeV does not interact strongly Electromagnetic decay µ+ → e+γ is forbidden Decays weakly – flavour changing − − µ → e ν e ν µ + + µ → e ν e ν µ Amplitude and Decay Rate 2 2 Matrix element / amplitude ∝ gWgW 1/(q –MW ) 2 2 2 q << MW → decay rate Γµ ∝ GF 2 5 dimensions → Γµ ∝ GF mµ Total decay rate 2 5 1 GF mµ = Γµ = 3 (h = c = 1) or lifetime τµ τ µ 192π Measurements -6 Lifetime τµ = 2.19703(4) 10 s Mass mµ = 105.658369(9) MeV -5 -2 Î GF = 1.16637(1) ·10 GeV weak coupling constant GF is measured in muon decay th see 4 year projects for measuring τµ and mµ Nuclear and Particle Physics Franz Muheim 7 LeptonLepton UniversalityUniversality inin WeakWeak InteractionInteraction Tau Decays mτ = 1.777 GeV > mµ, mπ, mρ, … Several weak decay modes possible Branching BF(τ − → X ) = Γ(τ − → X ) / Γ(τ − → all) − − Fractions τ → e ν eν τ BF = 17.8 ± 0.1% − − τ → µ ν µν τ BF = 17.4 ± 0.1% τ − → hadronsν BF = 64.7 ± 0.2% Tau Decay Rates τ − − Investigate decay τ → e ν eν τ 1 Γ 1 1 G 2 m 5 = Γ = τ →all Γ = Γ = F τ τ →all τ →eν eνυ τ →eν eνυ 3 ττ Γτ →eν ν BF 0.178 192π e υ τ →eνeνυ compare with muon decay m 5 τ = τ BF µ = ()2.91± 0.01 ×10−13 s Expect lifetime τ µ τ →eν eντ 5 mτ −13 Measure ττ = (2.906 ± 0.011)×10 s Î Weak coupling of τ and µ identical Lepton Universality in Standard Model W± boson couples identically to all leptons − − − + − − − + − − − + e → W ν e or ν e → e W µ → W ν µ or ν µ → µ W τ → W ντ or ντ → τ W Charged weak current ⎛ν ⎞ ⎛ν ⎞ ⎛ν ⎞ ⎜ e ⎟ ⎜ µ ⎟ ⎜ τ ⎟ ⎜ − ⎟ ⎜ − ⎟ ⎜ − ⎟ Couples within lepton doublets ⎝e ⎠ ⎝ µ ⎠ ⎝τ ⎠ Nuclear and Particle Physics Franz Muheim 8 WeakWeak InteractionInteraction ofof QuarksQuarks Weak charged quark currents Are weak charged quark currents also universal? -) - - e.g. is M(d → uW = M(µ → νµW ) ? Nature is not quite that simple Comparison Measure GF in muon and nuclear beta decay Obtain ratio GF(beta)/ GF(muon) = 0.974(3) Î Charged weak current almost equal for leptons and up/down quarks Leptonic Pion and Kaon Decays + + + + Observe π → µ νµ and K → µ νµ Evidence for flavour changing coupling u → sW+ between quark generations + + + + Rate Γ(K → µ νµ ) ≈ 5% expected Γ( π → µ νµ ) +) Matrix element M(u → sW ~ sinθC + M(u → dW ) ~ cosθC Nuclear and Particle Physics Franz Muheim 9 CabibboCabibbo MixingMixing AngleAngle Quark Flavour Mixing Two generations & four quark flavours (u, d, c, s) weak eigenstates not equal to mass eigenstates are linear combination of mass eigenstates Cabibbo Hypothesis ⎛ d'⎞ ⎛ cosθC sinθC ⎞ ⎛d ⎞ W boson couples to ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟ ⎝ s' ⎠ ⎝ − sinθC cosθC ⎠ ⎝ s ⎠ weak doublet (u,d’) Cabbibo angle θC rotates weak eigenstates Coupling strength to W boson within doublet ⎛ν ⎞ ⎛ u⎞ ⎛ u⎞ ⎜ e ⎟ → g ⎜ ⎟ → g cosθ = g V ⎜ ⎟ → g sinθ = g V ⎜ − ⎟ W ⎜ ⎟ W C W ud ⎜ ⎟ W C W us ⎝ e ⎠ ⎝ d ⎠ ⎝ s ⎠ Experimental Results − − 2 5 3 Muon decay Γ(µ → e ν eν µ ) = GF mµ /192π = Γµ 2 2 2 2 Beta decay Γ()d → uν eν µ ∝ GF cos θC = GF Vud + + 2 2 2 2 Pion decay Γ()π → µ ν µ ∝ GF cos θC = GF Vud + + 2 2 2 2 Kaon decay Γ()K → µ ν µ ∝ GF sin θC = GF Vus 0 Cabibbo angle θC = 12.96(9) sinθC = 0.2243(16) Cabibbo-Kobayashi-MaskawaCabibbo-Kobayashi-MaskawaWeak Charged Interaction Mechanism Mechanism W couples quarks of different generations For 3 generations ⎛ d'⎞ ⎛V V V ⎞ ⎛ d ⎞ ⎜ ⎟ ⎜ ud us ub ⎟ ⎜ ⎟ s' = V V V s expand to ⎜ ⎟ ⎜ cd cs cb ⎟ ⎜ ⎟ ⎜ b'⎟ ⎜V V V ⎟ ⎜ b ⎟ Nuclear and Particle Physics⎝ ⎠ Franz⎝ Muheimtd ts tb ⎠ ⎝ ⎠ 10 WeakWeak HadronHadron DecaysDecays Lightest Mesons and Baryons only flavour changing weak decay possible for Pions (π+ = u dbar), kaons (K+ = u sbar), charmed (D) and B mesons, neutron, Λ, Σ, Ξ, Ω Exceptions: zero net quantum numbers allow electromagnetic decay, e.g. π0 →γγ SpectatorSpectatorSpectator ModelModelModel Weak decay of heavy B or D meson (b or c quark) Decay rate of B or D meson equal to decay rate of heavy b or c quark, light quark is only spectator Predictions of Spectator Model 2 5 Semileptonic decay rate G m 2 Γ()B → X e −ν = F b V - c,u e 192π 3 cb(ub) B → Xc,u e νbar Equal total decay rates/lifetimes for + 0 + + 0 + D , D , Ds and B , B , Bs D meson lifetimes τ + ≈ 2.5τ 0 ≈ 2.5τ + D D Ds B mesons lifetimes are equal within 10% Spectator model works well for B mesons Nuclear and Particle Physics Franz Muheim 11.
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