Lecture: Standard Model of Particle Physics Heidelberg SS 23 Weak Interactions I Standard Model of Particle Physics SS 23
ors and Helicity States momentum vector in z direction u R = p, = / 2 u L = p, = / 2 v L = p, = / 2 v R = p, = / 2 fermions: (u) (u2) or more general: (v) (v2) u R = E m p E m u L = E m p E m ur vl anti-fermions: p vr = E m E m p E m vl = E m ( ) limit p ( ) 2 ul vr Standard Model of Particle Physics SS 23
Chirality Operator limit: m, p operator: right chirality 5 ur = u R 5 v L = v L 5 ul vr () ur ~ v L ~ 2 3 γ =iγ γ γ γ = ( ) = ( ) left chirality 5 u L = u L 5 v R = v R eigenvalues ± left-handed (chiral) particles: - right-handed (chiral) particles: note: a right-handed chiral anti-particle has a left-handed 3 Standard Model of Particle Physics SS 23
Projection Operator Definition: Π ± ±γ5 = 2 in the limit of E infinity fermions Π ur = ur ul = u L anti-fermions ul = ur = v L = vl vr = vr vr = vl = can reformulate Dirac Equation: μ i γ μ u R =m u L μ i γ μ u L =m u R note: massive fermions must have left-handed and right handed components 4 Standard Model of Particle Physics SS 23
Recap General Four Fermion Lagrangian G L= ( f Γ f ' ) ( f ' ' Γ f ' ' ' ) 2 Vector Current: scalar coupling: ψ λ=ψ j V = Axial-vector Current: pseudoscalar coupling: 5 j A = 5 γ ψ λ=ψ Tensor Coupling σ μaν = ψ ( γμ γ ν γ ν γμ ) ψ 5 Standard Model of Particle Physics SS 23
Vector, Axial and Scalar Currents Vector Current: μ μ j V = u γ u μ in QED: j V = μ μ u γ u = u L γ u L u R γ u R (no flip) Axial-vector Current: μ μ 5 ja = u γ γ u μ 5 note: μ 5 μ 5 u γ γ u = u L γ γ u L u R γ γ u R 5 = 5 (no flip) Scalar Coupling u u = u R u L u L u R ( flip!) Relations: j L = / 2 j V j A j R = / 2 j V j A 6 Standard Model of Particle Physics SS 23
Helicity Example Particle at almost rest ( < p ) with spin orientation in z direction: λ = j z = : j p H= = p z-axis N / 2 N / 2 = N / 2 N / 2 polarisation: 7 (classical) Standard Model of Particle Physics SS 23
Helicity Example Particle at almost rest ( < p ) with spin orientation in z direction: N / 2 N / 2 = N / 2 N / 2 polarisation: λ = j z = : j p H= = p z-axis (classical) In a classical approach states with defined helicities (H = ±) can be prepared. In Quantum mechanics the spin and momentum are to be replaced by operators. For massive particles the is not Lorentz invariant. This is easily seen in the above example: For p p' = -p the makes a flip: H H' = - 8 Standard Model of Particle Physics SS 23
Helicity Example Particle almost at rest ( < p ) with spin orientation in z direction: N / 2 N / 2 = N / 2 N / 2 polarisation: λ = j z = : j p H= = p z-axis (classical) In a Lorentz invariant theory, particle interactions can not be described by non-lorentz invariant quantities! The solution to this problem is provided by the Dirac equations Dirac spinors: only the chiral states in the limit p are Lorentz invariant u = ul ur ur = () 9 ul = () Standard Model of Particle Physics SS 23
Helicity Example Particle almost at rest ( < p ) with spin orientation in z direction: λ = j z = : j p H= = p z-axis u = u L u R = E m p E m E m p = E m ( ) ( ) N / 2 N / 2 = N / 2 N / 2 polarisation: (classical) () () 2m 2m Standard Model of Particle Physics SS 23
Helicity Example Particle at almost rest ( < p ) with spin orientation in z direction: λ = j z = : j p H= = p z-axis u = u L u R = E m p E m N / 2 N / 2 = N / 2 N / 2 polarisation: E m p = E m ( ) ( ) (classical) () () 2m 2m Projection of right state Π () () γ 5 2 m u= u= 2 2 2 m 2 Π () () γ 5 2 m u= u= 2 2 2 m 2 Standard Model of Particle Physics SS 23
Helicity Example Particle at almost rest ( < p ) with spin orientation in z direction: λ = j z = : j p H= = p z-axis u = u L u R = E m p E m N / 2 N / 2 = N / 2 N / 2 polarisation: E m p = E m ( ) ( ) (classical) () () 2m 2m Projection of right state Π () () γ 5 2 m u= u= 2 2 2 m 2 Π 5% right chiral states 2 () () γ 5 2 m u= u= 2 2 2 m 2 5% left chiral state Standard Model of Particle Physics SS 23
Test of Lorentz Structure? Fermi transition LH anti-neutrino Gamov Teller transition vector axialvector? * n scalar - - n p - - p tensor * * n? - p - - * n 3 - p - - Standard Model of Particle Physics SS 23
Test of Lorentz Structure? Fermi transition RH anti-neutrino Gamov Teller transition vector axialvector? * n scalar - - n p - - p tensor * * n? - p * n 4 - p Standard Model of Particle Physics SS 23
Test of Lorentz Structure? Fermi transition RH anti-neutrino Gamov Teller transition vector axialvector? * - n p - * n scalar measure correlations? and directions: - * - n p * n 5 tensor of particle spins - p polarisations momenta(directions energies) - p Standard Model of Particle Physics SS 23
Important Experiments Wu-Experiment (957): radioactive decay of Co6 Goldhaber-Experiment (958): radioactive decay of Eu52 Muon Decay: Michel spectrum Pion Decay: branching ratios Neutrino Nucleon Scattering: neutrino-antineutrino Nuclear Beta Decays 6 Standard Model of Particle Physics SS 23
Idea of the Wu-Experiment 6 Co 6Ni e- ν- High of Cobalt leads to Gamov-Teller transition Polarisation of electron (neutrino)? λ (e )=? 7 Standard Model of Particle Physics SS 23
Test of Lorentz Structure in Co 6 Gamov Teller transitions axialvector? axialvector? 6 Co 6 - Ni tensor 6 Co 6 Co 6 - Ni tensor 6 Ni - - 8 6 Co - 6 Ni Standard Model of Particle Physics SS 23
Measurement of the Ni*-Polarisation important cross check! 6 Ni*(J=4) is produced in an excited state! Beta Decay followed by photo-nuclear decay: 6 Ni* 6Ni* γ γ γ 9 Photons are polarised and oriented (symmetrically) in direction of the Ni (Co) polarisation axis. Maximum is orthogonal to Ni (Co) polarisation Standard Model of Particle Physics SS 23
light guide a) equatorial counter b) polar counter pump nozzle (helium) b) photon detection Co 6 paramagnetic container T=. K anthracen crystal (beta detector) a) photon detection 2 Standard Model of Particle Physics SS 23
Conclusion Wu-Experiment Electrons are dominantly emitted opposite to Co spin direction Product J p / p () is non-zero! Helicity is negative! Discrete-Parity symmetry is violated (initial state had H=) Note: the angular distribution of the electrons is given by: dn A cos θ d cos θ 2 Standard Model of Particle Physics SS 23
Test of Lorentz Structure in Co 6 Gamov Teller transitions axialvector? axialvector? 6 Co 6 - Ni tensor 6 Co 6 Co 6 - Ni tensor 6 Ni - - 22 6 Co - 6 Ni Standard Model of Particle Physics SS 23
Test of Lorentz Structure in Co 6 Gamov Teller transitions axialvector? 6 Co 6 - Ni? tensor What about the neutrino 6 Co 6 Ni - - 23 Standard Model of Particle Physics SS 23
Europium Decay Chain Lifetime 52Eu = 3.5 a 52 Eu e- 52Sm* ν Europium has no nuclear spin Polarisation of neutrino and Sm* are opposite! The neutrino polarisation can be measured by determining the Sm* polarisation Eu e- Luckily Sm* decays further: Sm* 52Sm γ Sm* ν photon is of low energy if emitted opposite to Sm* flight direction fluorescence exploit Compton scattering to determine photon polarisation Sm fluorescence 24 Standard Model of Particle Physics SS 23
Goldhaber Experiment Compton scattering scatterer magnetised iron shield resonance fluorescence 25 Standard Model of Particle Physics SS 23
Goldhaber: Result Conclusion Due to the ingenious construction of the experiment the polarisation of the photon corresponds to the polarisation of the neutrino The photon was measured to be left-handed! As a result the of the neutrino has to be left handed Left-handed chiral (neutrino!) fermion couplings can be described either by V-A coupling or T(P)-S couplings. T(P)-S couplings excluded by Wu- and Goldhaber-experiments 26 Standard Model of Particle Physics SS 23
Test of Lorentz Structure in Co 6 Gamov Teller transitions axialvector? Goldhaber,Wu axialvector? 6 Co 6 Left-handed - Ni tensor 6 Co 6 Co 6 - Ni tensor Goldhaber - 6 Ni Wu - - 27 6 Co - 6 Ni Standard Model of Particle Physics SS 23
Mott Scattering for Polarisation Measurements reaction: e N e N is polarisation dependent: coupling of spin and orbital momentum left-right asymmetry 28 Standard Model of Particle Physics SS 23
Polarisation in Beta Decays polarisation: N N 2 2 P = N N 2 2 in different radioactive decays: P v /c BILD only left-handed particles! 29 Standard Model of Particle Physics SS 23
Myon Decay classical 3-body decay μ e ν e νμ only left handed particles interact G L= ( f Γ f ' ) ( f ' ' Γ f ' ' ' ) with 2 3 Γ μ=γ μ ( γ 5 ) Standard Model of Particle Physics SS 23
Lorentz Structure in Muon Decay Fermi transition Gamov Teller transition vector axialvector * - μ- - e νμ μ - μ- νμ - Momentum Momentum * * spin momentum - νμ - e 3 * - μ- νμ - Standard Model of Particle Physics SS 23
Lorentz Structure in Muon Decay Fermi transition Gamov Teller transition vector axialvector * μ- - νμ - e myon neutrino has highest energy * - μ- νμ - electron has highest energy From considerations, the electron is expected to have in average an energy of 3/4 of half the muon mass 3/8 (exact: 7/) in contrast to /3 naively expected from kinematics Michel spectrum 32 Standard Model of Particle Physics SS 23
Michel-Spectrum e e Very good agreement between measurement and SM prediction (V-A) theory 33 Standard Model of Particle Physics SS 23
Michel-Spectrum beyond the SM The muon decay is one of the best SM testing grounds: Extended Michel spectrum formula: x = 2 me mμ 2 2 m e mμ Inportant parameters: They can be related to 4-fermion couplings Experimental results: G L= ( f Γ f ' ) ( f ' ' Γ f ' ' ' ) 2 Confirmation of V-A coupling 34 Standard Model of Particle Physics SS 23
Test Charged Pion Branching Ratios dominant decay: B(π μ ν) = 99.9877 % suppressed decay: B(π e ν) =.23-4 Similar to the neutral pion in QED the more obvious decay is suppressed! V-A currents conserve. V-A Currents: Resulting spin should be J(π)= jμweak = v (x) γ μ ( γ 5 )u( x) μ But pion is a Pseudo-scalar J(π)= suppression W π Decay width: ν G 2F 2 m2μ 2 Γ(π μ )= f m m ( 2 ) 8π π π μ mπ Polarisation of state is given by fermion velocity: λ = β m2e ( m2e /m2π ) Γ(π e ) 4 = Γ(π μ ) m2μ ( m2μ / m2π ) for left chiral states 35 Standard Model of Particle Physics SS 23
Measurement of π e ν (Britton PRL 68, 2, 992, 3) 3-body 2-body Energy Timing Pion lifetime 25 ns Muon lifetime 2 μs Result: B(π e ν) =.23-4 Conclusion: no scalar coupling! 36 Standard Model of Particle Physics SS 23
Gamma Matrices II = = i 2 i = i i 5 = 3 = (in other representations g5 is diagonal ) 37 Standard Model of Particle Physics SS 23
38 Standard Model of Particle Physics SS 23