Lecture: Standard Model of Particle Physics Heidelberg SS 23 Fermi Theory Standard Model of Particle Physics SS 23
2 Standard Model of Particle Physics SS 23
Weak Force Decay of strange particles Nuclear beta decay (+fission) 3 Standard Model of Particle Physics SS 23
Fermi Theory Unified description of all kind of beta decays? nuclear decays muon and pion decay decay of strange hadrons and heavy quarks Description of weak scattering processes? at low energy at high energy 4 Standard Model of Particle Physics SS 23
Recap I Lagrangian (independent of energy) G L= ( f Γ f ' ) ( f ' ' Γ f ' ' ' ) 2 most general ansatz for operator Γ: 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
Recap II Vector Current conservation μ μ μ (no helicity flip) u γ u = u L γ u L + u R γ u R Scalar Coupling u u = u R u L + u L u R (helicity flip!) 6 Standard Model of Particle Physics SS 23
Recap II Vector Current conservation μ μ μ (no helicity flip) u γ u = u L γ u L + u R γ u R Scalar Coupling u u = u R u L + u L u R (helicity flip!) What is the difference between helicity and chirality? 7 Standard Model of Particle Physics SS 23
Recap II Vector Current conservation μ μ μ (no helicity flip) u γ u = u L γ u L + u R γ u R Scalar Coupling u u = u R u L + u L u R (helicity flip!) What is the difference between helicity and chirality? What is the difference between helicity and polarisation? 8 Standard Model of Particle Physics SS 23
Recap II Vector Current conservation μ μ μ (no helicity flip) u γ u = u L γ u L + u R γ u R Scalar Coupling u u = u R u L + u L u R (helicity flip!) What is the difference between helicity and chirality? What is the difference between helicity and polarisation? What is the difference between helicity flip and spin flip? 9 Standard Model of Particle Physics SS 23
Recap III Energy dependence of electromagnetic interaction: s Standard Model of Particle Physics SS 23
3-Body Decay e.g.: n p e ν p e electron energy: E(e) neutrino energy: E(ν) ν Fermi's Golden rule: 2 dn π dn ( p) dp = 2 f H i ℏ de with: 2 2 H fi = f H i =const 2 dn V 2 2 = p dp p dp ν ν e e Phase space: de 4 4 de 4π ℏ Beta Spectrum: 2 2 2 dn (η) d η = H fi η (ϵ ϵ) d η with: η = pe / m e ϵ = E e / me Standard Model of Particle Physics SS 23
Kurie-Plot (Fermi diagram) dn (η) 2 2 H fi η = ϵ ϵ Beta decay of 6He 6Li e- ν Note that the neutrino mass was set to zero here! linear function! matrix element independent of energy! Fermi Theory 2 = constant 2 H fi Standard Model of Particle Physics SS 23
Mass and Resolution Effects finite neutrino mass limited energy resolution: 3 Standard Model of Particle Physics SS 23
Neutrino-Mass Measurement 4 Standard Model of Particle Physics SS 23
Katrin Experiment Measure Beta-Spectrum in the tritium decay current limit: mν< ev expected limit: mν<~.-.3 ev 5 Standard Model of Particle Physics SS 23
Lifetime in Beta Decay Transition probability depends only on available decay energy E from beta Spectrum: 2 2 2 dn (η) d η = H fi η (ϵ ϵ) d η with η ϵ Decay width 5 ϵ M 5 Lifetime depends on the fifth power of the particle mass (muon decay) Q value (nuclear decay Q ~ Ee + Eν) 6 (only little recoil) Standard Model of Particle Physics SS 23
Weak Force Lifetimes in weak decays of order - seconds years Interaction length: Nuclear interaction (Fe): λstrong ~ O() cm Weak interaction (Fe): >> 3 km (neutrino energy dependent) 7 Standard Model of Particle Physics SS 23
Weak Force Discovery of muon neutrino (Lederman et al.): only tiny fraction of neutrinos is detected The weak force is really weak! 8 Standard Model of Particle Physics SS 23
Weak scattering processes A constant matrix element (Fermi theory) gives an energy dependent cross section in weak scattering processes Reason: phase space! e νe νμ μ σ (νμ e μ ν e ) G 2F s neutrino-electron scattering d u νμ μ σ ( νμ d μ u) G 2F s neutrino-nucleon scattering 9 Standard Model of Particle Physics SS 23
Weak scattering processes I Kinematics: Fixed Target: s=2 E ν M target Scattering cross section should rise linearly with neutrino beam energy linear rise cross section (Fermi theory) beam energy 2 Standard Model of Particle Physics SS 23
Weak scattering processes II Center of mass energies limited in Fixed Target Experiments Trick: invert reaction at colliders: ν e d e u e+ d ν e u Kinematics: target Fixed Target: s=2 E ν Collider: s=4 E e E p coll M target (HERA: electron-proton) from comparison: E target ν 2 E coll e Ep 5 TeV M target 2 Standard Model of Particle Physics SS 23
Electron-Proton Collider HERA Ee = 26.7 GeV Ep= 92 GeV 22 Standard Model of Particle Physics SS 23
Lorentz Invariant Kinematics of Deep Inelastic Scattering Process The virtuality of the exchanged photon is given by: p' θ Q 2 = q 2 = ( p p ') 2 p positron 4 sin / 2 q Relative energy loss (inelasticity): y = ν E Elektron neutrino qp = pp W-boson xp P proton relative fraction of parton momentum: q2 Q2 x = = 2q P sy with cms energy: s = 2pP 23 Standard Model of Particle Physics SS 23
Charged Current Event at H neutrino hadronic jet proton (92 GeV) positron (26.7 GeV) neutrino + e d ν e u 24 Standard Model of Particle Physics SS 23
Weak scattering processes III HERA beam energy translated into Fixed target linear rise cross section 992 (3 events) (Fermi theory) H effect of W-propagator GF g 2 2 M W + Q2 beam energy Breakdown of Fermi theory at high energies s/2 ~ GeV 25 Standard Model of Particle Physics SS 23
Lorentz Structure of Weak Process? Lagrangian (independent of energy) G L= ( f Γ f ' ) ( f ' ' Γ f ' ' ' ) 2 most general ansatz for operator Γ: Vector Current: j V = scalar coupling: ψ λ=ψ (Fermi s proposal): 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 26 Standard Model of Particle Physics SS 23
Test of Lorentz Structure? Different transitions in weak decays: n p e ν Fermi transition Gamov Teller transition +/2 p spin +/2 n e ν no spin flip (ΔJ=) electron and neutrino in singlet state 27 -/2 p { spin +/2 n + { + + e ν spin flip (ΔJ=,) electron and neutrino in triplet state Standard Model of Particle Physics SS 23
Test of Lorentz Structure? Different transitions in weak decays: n p e ν Fermi transition Gamov Teller transition +/2 p spin +/2 n e ν no spin flip (ΔJ=) electron and neutrino in singlet state Note: -/2 p { spin +/2 n + { + + e ν spin flip (ΔJ=,) electron and neutrino in triplet state spin flip helicity flip 28 Standard Model of Particle Physics SS 23
Test of Lorentz Structure? Fermi transition fix electron helicity= - vector Momentum coupling axialvector Momentum coupling? + Spin helicity * n scalar Momentum coupling - helicity e- ν- n p - - + e- ν- p tensor Momentum coupling + - * n? * + Spin - + p Spin helicity Gamov Teller transition + Spin helicity - - e- ν- * n 29 - p - - e- ν- Standard Model of Particle Physics SS 23
Test of Lorentz Structure? Fermi transition RH anti-neutrino helicity Gamov Teller transition vector Momentum coupling axialvector Momentum coupling? + Spin helicity * n scalar Momentum coupling - e- ν- - n p - - + e- ν- p tensor Momentum coupling + * * n? Spin helicity helicity - + p + Spin + Spin helicity + + e- ν- * n 3 - p + + e- ν- Standard Model of Particle Physics SS 23
Test of Lorentz Structure? Fermi transition fix electron helicity= - vector Momentum coupling axialvector Momentum coupling? + Spin helicity * n Gamov Teller transition - p + Spin helicity - + e- ν- * n - p - + e- ν- Both, Fermi and Gamov Teller transitions observed in weak interactions Vector and Axialvector currents conserve helicities and realised in nature 3 Standard Model of Particle Physics SS 23
Test of Lorentz Structure? Scalar and tensor (pseudoscalar) couplings flip helicity! Have to measure spin-orientation of decay leptons! scalar Momentum coupling? + Spin helicity tensor Momentum coupling * - n p + Spin helicity - - e- ν- * n 32 - p - - e- ν- Standard Model of Particle Physics SS 23
The End 33 Standard Model of Particle Physics SS 23
Gamma Matrices II = = i 2 i = i i 5 = 3 = (in other representations g5 is diagonal ) 34 Standard Model of Particle Physics SS 23
Spinors of Helicity States u R = p, = / 2 u L = p, = / 2 v L = p, = / 2 v R = p, = / 2 fermions: (u) (u2) (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 ( ) 35 ul vr Standard Model of Particle Physics SS 23
Chirality Operator limit m ur ~ v L ~ operator: 5 = i 2 3 = right chirality left chirality 5 ur = u R 5 v L = v L ul ~ vr ~ 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 helicity 36 Standard Model of Particle Physics SS 23
Projection Operator definition: Π ± ± γ5 = 2 in the limit of E infinity fermions + anti-fermions Π ur = ur ul = u L ul = ur = v L = vl vr = vr vr = vl = projection: =R =L (right-handed (chiral) state) (left-handed (chiral) state) reformulate Dirac Equation: i R =m L i L =m R note: massive fermions must have left-handed and right handed components 37 Standard Model of Particle Physics SS 23
Vector and Axial Currents Vector Current: j V = R R, L L in QED: j V = (conservation of currents) Axial-vector Current: 5 j A = note: 5 = 5 Left (right)-handed Current: j L = j R = relations: j L = / 2 j V j A weak interaction (V-A theory): j R = / 2 j V j A 38 Standard Model of Particle Physics SS 23
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