Presentation on theme: "Feynman Diagrams Feynman diagrams are pictorial representations of"— Presentation transcript:
1 Feynman Diagrams Feynman diagrams are pictorial representations of P Spring 2003 L3Feynman DiagramsRichard KassFeynman diagrams are pictorial representations ofAMPLTUDES of particle reactions, i.e scatterings or decays. Use of Feynman diagrams can greatly reduce the amount of computation involved in calculating a rate orcross section of a physical process, e.g.muon decay: m-®e-nenm or e+e-® m+m- scattering.Feynman and his diagramsLike electrical circuit diagrams, every line in the diagram has a strict mathematical interpretation. Unfortunately the mathematical overhead necessary to do complete calculations with this technique is large and there is not enough time in this course to go through all the details. The details of Feynman diagrams are addressed in Advanced Quantum and/or For a taste and summary of the rules look at Griffiths (e.g. sections 6.3, 6.6, and 7.5) or Relativistic Quantum Mechanics by Bjorken & Drell.
2 Feynman Diagrams Each Feynman diagram represents an AMPLITUDE (M). P Spring 2003 L3Richard KassFeynman DiagramsEach Feynman diagram represents an AMPLITUDE (M).Quantities such as cross sections and decay rates (lifetimes) are proportional to |M|2.The transition rate for a process can be calculated using time dependent perturbationtheory using Fermi’s Golden Rule:In lowest order perturbationtheory M is the fourier transformof the potential. M&S B.20-22, p295“Born Approximation” M&S 1.27, p17The differential cross section for two body scattering (e.g. pp®pp) in the CM frame is:qf=final state momentumvf= speed of final state particlevi= speed of initial state particleM&S B.29, p296The decay rate (G) for a two body decay (e.g. K0® p+p-) in CM is given by:Griffiths 6.32m=mass of parentp=momentum of decay particleS=statistical factor (fermions/bosons)In most cases |M|2 cannot be calculated exactly.Often M is expanded in a power series.Feynman diagrams represent terms in the series expansion of M.
3 Feynman Diagrams OR Feynman diagrams plot time vs space: P Spring 2003 L3Feynman DiagramsRichard KassFeynman diagrams plot time vs space:Moller Scattering e-e-®e-e-e-M&Sstylee-QED RulesSolid lines are charged fermionselectrons or positrons (spinor wavefunctions)Wavy (or dashed) lines are photonsArrow on solid line signifies e- or e+e- arrow in same direction as timee+ arrow opposite direction as timeAt each vertex there is a coupling constantÖa, a= 1/137=fine structure constantQuantum numbers are conserved at a vertexe.g. electric charge, lepton number“Virtual” Particles do not conserve E, pvirtual particles are internal to diagram(s)for g’s: E2-p2¹0 (off “mass shell”)in all calculations we integrate over the virtualparticles 4-momentum (4d integral)Photons couple to electric chargeno photons only verticesspaceinitial statefinal statee-e-ORtimefinal statetimeinitial statespace
4 P Spring 2003 L3Richard KassFeynman DiagramsWe classify diagrams by the order of the coupling constant:Bhabha scattering: e+e-®e+e-Amplitude is of order a.a1/2a1/2a1/2a1/2a1/2Amplitude is of order a2.a1/2Since aQED =1/137 higher order diagrams should be correctionsto lower order diagrams.This is just perturbation Theory!!This expansion in the coupling constant works for QED since aQED =1/137Does not work well for QCD where aQCD »1
5 P Spring 2003 L3Richard KassFeynman DiagramsFor a given order of the coupling constant there can be many diagramsBhabha scattering: e+e-®e+e-amplitudes caninterfere constructivelyor destructivelyMust add/subtract diagram together to get the total amplitudetotal amplitude must reflect the symmetry of the processe+e-®gg identical bosons in final state, amplitude symmetric under exchange of g1, g2.g1e-g1e-+g2g2e+e+Moller scattering: e-e-®e-e- identical fermions in initial and final stateamplitude anti-symmetric under exchange of (1,2) and (a,b)e-1e-1e-ae-b-e-2e-be-2e-a
6 P Spring 2003 L3Richard KassFeynman DiagramsFeynman diagrams of a given order are related to each other!e+e-®ggg’s in final stategg®e+e-g’s in inital statecompton scatteringg e-®g e-electron and positronwavefunctions are relatedto each other.
7 Anomalous Magnetic Moment P Spring 2003 L3Anomalous Magnetic MomentRichard KassCalculation of Anomalous Magnetic Momentof electron is one of the great triumphs of QED.In Dirac’s theory a point like spin 1/2 object of electric charge q and mass m has a magnetic moment: m=qS/m.In 1948 Schwinger calculated the first-order radiative correctionto the naïve Dirac magnetic moment of the electron.The radiation and re-absorption of a single virtual photon,contributes an anomalous magnetic moment of ae = a/2p »Basic interactionwith B field photonFirst order correctionDirac MM modified to: m=g(qS/m) and deviation from Dirac-ness is: ae º (ge - 2) / 23rd order correctionsThe electron's anomalous magnetic moment (ae) is now known to 4 parts per billion.Current theoretical limit is due to 4th order corrections (> dimensional integrals)The muons's anomalous magnetic moment (am) is known to 1.3 parts per million.(recent screwup with sign convention on theoretical am calculations caused a stir!)