Lecture 11 - Weak Interactions Weak Charged Currents and β Decay • Weak Neutral Currents • Lepton-Neutrino Scattering • Muon and Tau Decays • Lepton Universality • Lepton Flavour Conservation • 1 Weak Charged Currents The exchange of a heavy W + or W − boson describes weak interactions with charged currents The fermion currents are either: A charged lepton changing into a neutrino (or vice-versa) • These must be of the same flavour! An up-type quark changing into a down-type quark • Weak interaction strength is related to the Fermi constant GF : 2 GF g −5 −2 = 2 GF = 1.16 10 GeV √2 8MW × where g is a dimensionless coupling constant 2 Description of β Decay Quark level: + − u de νe d ue ν¯e → → Hadron level (mp = 938MeV, mn = 940MeV): − n pe ν¯e is allowed → Free neutron is unstable and decays with τn = 886s + p ne νe is forbidden → Free proton is stable; decay forbidden by mp < mn Nuclear level: 14 14 ∗ + + O N + e + νe β decay → 60 60 ∗ − − Co Ni + e +ν ¯e β decay → Which type is allowed depends on energy available between nuclei 3 Feynman diagram for β Decay ν¯e W − d e− ¡ u 2 2 Propagator for W boson is igµν /(M q ) • − W − Dimensionless coupling constant is g for leptons • modified by CKM factors for quarks (Vud) The charged current operator is γµ 1 (1 γ5) instead of γµ • 2 − Vector minus Axial-vector (V-A) 4 Amplitude for β Decay The matrix element “factorizes” into lepton and quark currents: g µ 1 5 1 g µ 1 5 = u¯dγ (1 γ )uu u¯ν γ (1 γ )ue M √2 2 − M 2 q2 √2 e 2 − W − Neglecting lepton masses the lepton current gives: µ 1 5 2 u¯ν γ (1 γ )ue = 8EeEν (1 + cos θ) | e 2 − | where θ is the opening angle between e and ν µ 1 5 2 The quark current squared u¯dγ (1 γ )uu is parametrised by a | 2 − | “hadronic form factor”, e.g. F (n p), which is a function of q2 → In nuclear β decay use “matrix elements”, e.g. 60Co 60 Ni∗ → 2 2 2 At low q can replace g /MW with Fermi constant GF 5 Tritium β decay (Kurie plot) 2 dΓ GF 2 2 Spectrum: = 3 E (E0 Ee) dEe π e 2 5 − 1 GF E0 Total rate: Γ = τ = 30π3 neglecting lepton mass and nuclear matrix element Shape near endpoint E0 determines m(νe) < 2eV 6 Weak Neutral Currents Described by the exchange of a heavy Z0 boson Z0 couples to all types of fermion but there are no flavour-changing neutral currents! Neutral current amplitude for νe νe scattering: → G µ 1 5 µ e e 5 = u¯ν γ (1 γ )uν u¯eγ (cV cAγ )ue M √2 2 − − cV and cA are vector and axial-vector couplings which depend on fermion type: Lepton 2cV 2cA Quark 2cV 2cA νe, νµ, ντ 1 1 u,c,t 0.38 1 e, µ, τ -0.06 -1 d,s,b -0.68 -1 7 Electron Neutrino Scattering Neutrino-Electron scattering ν e− ν e−: − − e e •e e → − e νe Z W ¡ νe νe ¡ νe e− Antineutrino-Electron scatteringν ¯ e− ν¯ e−: − − e e • →− − e e e e Z W ¡ ¡ ν¯e ν¯e ν¯e ν¯e 8 Muon Neutrino Scattering − − Muon neutrino-Electron scattering νµe νµe : • − − → e e e− e− Z Z ¡ νµ νµ ¡ ν¯µ ν¯µ − − Inverse muon decay νµe νeµ : − • e → νe W ¡ νµ µ− 9 νµ scattering cross-sections − − Charged current exchange diagram νµe νeµ : • → dσ G2 s G2 s 2 = 16G2 s2 = F σ = F |M| F dΩ 4π2 π Neutral current scattering of muon neutrinos: • 2 dσ GF s 2 2 2 = (cV + cA) +(cV cA) (1 y) dy 4π − − 2 − − GF s 2 2 σ(νµe νµe )= (c + cV cA + c ) → 3π V A Neutral current scattering of muon antineutrinos: • 2 − − GF s 2 2 σ(¯νµe ν¯µe )= (c cV cA + c ) → 3π V − A 10 νe scattering cross-sections νe scattering is superposition of charged and neutral currents Scattering of electron neutrinos: • 2 dσ − − GF s 2 2 2 (νee νee )= g + g (1 y) dy → π L R − Scattering of electron antineutrinos: • 2 dσ − − GF s 2 2 2 (¯νee ν¯ee )= g + g (1 y) dy → π R L − where gL =1+(cV + cA)/2 and gR =(cV cA)/2 − Total cross-sections: • 2 2 − G s − G s σ(ν e )= F (g2 + g2 /3) σ(¯ν e )= F (g2 + g2 /3) e π L R e π R L − −41 2 In the lab frame σ(νee ) 10 Eν cm where Eν is in GeV ≈ 11 Muon Decay ν¯e W − g µ− e g ¡ νµ Matrix element: GF µ 1 5 µ 1 5 = u¯ν γ (1 γ )uµ u¯eγ (1 γ )uν M √2 µ 2 − 2 − e Michel spectrum: 2 dΓ GF 2 2 4Ee = 3 mµEe 3 dEe 12π − mµ Total decay rate: 2 5 GF mµ 1 Γµ = 3 τµ = = 2.197µs 192π Γµ 12 Tau Decays The τ lepton is heavy enough to decay to many final states (mτ = 1.777 GeV) − − − − τ e ντ ν¯e τ µ ντ ν¯µ → → − − τ duν¯ τ τ suν¯ τ → → Naive branching fractions (with factor 3 for quark color): × (τ e)= (τ µ) = 20% (τ hadrons) = 60% B → B → B → The decay to an electron can be related to muon decay: 5 Γ → m τ e = τ Γµ mµ This agrees with the measured τ lifetime ττ = 291fs 13 Lepton Universality Couplings of W and Z bosons are the same for all families of leptons This can be tested in τ decays: − − (τ e ν¯eντ ) = (17.84 0.06)% B → ± − − (τ µ ν¯µντ ) = (17.36 0.06)% B → ± small difference is due to the muon mass Other tests include: Michel spectrum shape in µ and τ decays W ℓν and Z ℓ+ℓ− at high energy colliders → → Heavy quark decays b cℓν and c sℓν → → 14 50 years of µ e flavour violation searches → 15 25 years of τ flavour violation searches 16.