QED lecture10

Bernd Surrow Bernd Surrow Introduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 1 Introduction to Nuclear and Particle PhysicsIntroduction to Nuclear and Particle Physics8.7018.701Lecture 10Lecture 10QED IIQED II 2 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 EM EM basicsbasicsThe QED Feynman rulesThe QED Feynman rules OutlineOutline QED example processesQED example processesA) Electron-Muon scatteringA) Electron-Muon scatteringB) Electron-electron scattering (Moeller scattering)B) Electron-electron scattering (Moeller scattering)C) Electron-positron scattering (Bhabha scattering)C) Electron-positron scattering (Bhabha scattering)D) Photon-electron scattering (Compton scattering)D) Photon-electron scattering (Compton scattering)E) Electron-positron (Fermion pair production)E) Electron-positron (Fermion pair production)Casimir Casimir trick trick and and trace theoremstrace theoremsRenormalizationRenormalization 3 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 EM basicsEM basicsReviewMaxwell equations Field strength tensor: Anti-symmetryAnti-symmetry 2nd rank tensor in terms of E and B Formulation of inhomogeneous Maxwell equations: Four-vector: Continuity equation: (1)(2)(3)(4) 4 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 EM basicsEM basicsScalar and vector potentialThe 3rd Maxwell is equivalent to the statement for the magnetic field:The 2nd Maxwell takes the following form:The electric field can be written as follows:Relativistic notation: Defect of potential formulation:⇒ V and A are not uniquely determined!New possible potential:Change of potential: Gaugetransformation: Lorentz condition 5 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 EM basicsEM basicsPhoton fieldInhomogeneous Maxwell equationsWith Lorentz condition:Free photon case:Solution: Constrain: Massless particle: Photon Polarization vectorPhoton field in QED 6 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 EM basicsEM basicsPolarization vectorLorentz condition requires:In the Coulomb gauge, i.e.:The polarization three-vector is perpendicular to the direction of propagation: Freephoton is transversely polarized: Coulomb gauge = Transverse gaugeTwo linear independent three-vectors:Note:•Massless particle: Only two spin states (helicity +1 or -1) regardless of its spin, exceptspin 0 with only one spin state!•Massive particle: 2s+1 spin states (3 for spin 1 particle!) Two linear independent three-vectors perpendicular to p! 7 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 QED Feynman rulesQED Feynman rulesLagrangian in QEDSum of Dirac free field (charged particles) and interaction term:Compare this to:Local gauge invariance•LQED can be formally obtained from Lfree with the following substitution:•With this substitution, Lfree is invariant under the following local gauge transformation: U(1)•With the above substitution of the partial derivative, we convert a global invariant Lagrangian intoa locally invariant Lagrangian. This is called minimal coupling rule! Here: 8 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 QED Feynman rulesQED Feynman rulesGeneral considerations: Electron/PositronRelativistic quantitiesElectron and positron spinorDirac equation for electron and positronsProperties among electron and positron spinors•Orthogonal:•Normalization:•Completeness: Adjoint: 9 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 QED Feynman rulesQED Feynman rulesGeneral considerations: PhotonRelativistic quantitiesPhoton fieldLorentz gaugeCoulomb gauge 10 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 QED Feynman rulesQED Feynman rules1 NotationLabel incoming and outgoing four-momenta: p1, p2,…, pn and the corresponding spins: s1, s2,…, snAssign arrows:•External lines: Electron, positron•Internal lines: Preserve direction of flow2 External linesElectron:Incoming:Outgoing:Positron:Incoming:Outgoing:Photon:Incoming:Outgoing: 3 Vertex factor4 PropagatorElectron/PositronPhoton 11 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 QED Feynman rulesQED Feynman rules5 Conservation of energy and momenta6 Integrate over internal lines 7 Cancel the delta functionThe result will include a factor:Corresponding to overall energy-momentum conservation. Cancel this factor and what remains is -iM!8 AntisymmetrizationInclude a minus sign between diagrams that differ only in the interchange of two incoming (or outgoing)electrons (or positrons), or of an incoming electron with an outgoing positron (or vice versa). 12 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 QED example processesQED example processesA) Electron-muon scatteringFeynman graphApplication of Feynman rules: Integration over q yields: 13 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 QED example processesQED example processesB) Electron-electron scattering (Moeller scattering) Interchange: p3,s3 p4,s4 14 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 QED example processesQED example processesB) Electron-electron scattering (Moeller scattering)Cross-section calculation Relativistic limit:Non-relativistic limit: Rutherford term!Rutherford term! Scaling in: 15 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 QED example processesQED example processesC) Electron-Positron scattering (Bhabha scattering) 16 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 QED example processesQED example processesC) Electron-Positron scattering (Bhabha scattering)Comparison of Bhabha scattering results at PETRA (DESY) and QED calculations Relativistic limit:Non-relativistic limit: Scaling in: Rutherford term!TASSOMARK-J 17 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 QED example processesQED example processesD) Photon-electron scattering (Compton scattering) Amplitudes: 18 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 QED example processesQED example processesD) Photon-electron scattering (Compton scattering) Hofstadter 1949 andBernstein 1956 19 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 QED example processesQED example processesE) Electron-positron (Fermion-pair production) Charge of fermion pair! Number of colors: : Nc=3 20 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 Casimir Casimir trick and trace theoremstrick and trace theoremsCasmir trickSee Latex documentResult: No spinors on left side! 21 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 Casimir Casimir trick and trace theoremstrick and trace theoremsTrace theoremsGeneral identities: Important relations among gamma matrices: 22 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 Casimir Casimir trick and trace theoremstrick and trace theoremsTrace theorems Note: The trace of the product of an odd number ofgamma matrices is zero! 23 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 Casimir Casimir trick and trace theoremstrick and trace theoremsTrace theorems involving γ5 matrix Note: The trace of the product of an odd number of gamma matricesmultiplied by a γ5 matrix is zero! 24 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 Casimir Casimir trick and trace theoremstrick and trace theoremsContraction of ε tensor 25 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 RenormalizationRenormalizationIntroduction Leading order:Same process involving loop diagram: Vacuum polarization 4 internal lines! Divergent integral! 26 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 RenormalizationRenormalizationDealing with divergent terms (1)The solution to this problem contributed enormously to the development of QED (Dirac,Pauli, Kramers, Weisskopf, Bethe besides Tomonaga, Schwinger and Feynman)The procedure to deal with divergent loop terms is call renormalization!General concept: Redefinition of masses and coupling constants Contains divergent terms!What we measure!Special case: Lowest order 27 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 RenormalizationRenormalizationDealing with divergent terms (2)Technical idea:With this procedure M becomes: Introduce renormalized coupling constant: 28 Bernd SurrowIntroduction to Nuclear and Particle Physics - 8.701Department of Physics, MIT - Spring 2007 RenormalizationRenormalizationDealing with divergent terms (3)In terms of gR we find then for M:Two terms:•Infinite term: absorbed in gR•Finite term: depends on q2 (f-term)Express end result in terms of α: Consequence of loopdiagram: q2 dependenceEffect in QED is small!Note: If all infinities arising fromhigher-order diagrams can beaccommodated throughrenormalized quantities, a theory issaid to be renormalizable!All gauge theories arerenormalizable: t’Hooft/Feldman