Feynman Diagrams For Pedestrians - Herbstschule Maria Laach

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Contents1 Introduction 11.1 Scattering Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Lorentz Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Asymptotic States 42.1 Klein-Gordon Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Free Spin-0 Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Anti Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Dirac Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.5 Gamma Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.6 Free Spin-1/2 Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.7 Free Spin-1 Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Interactions 153.1 Propagators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Feynman Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3 Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.4 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.5 Phase Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 QED 244.1 e + e − → µ + µ − . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.2 Trace Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.3 Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.4 FORM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.5 Bhabha Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.6 FORM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 QCD 315.1 Feynman Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.2 3-Jet Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 Standard Model 356.1 Propagators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356.2 Feynman Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366.3 Higgs Strahlung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 Solutions 39Author: ohlRevision: 172Date: 2012-08-27 16:30:28 +0200 (Mon, 27 Aug 2012)
1 Introduction1.1 Scattering AmplitudesBasic principle of quantum mechanics• an accelerator prepares an initial state |in〉• that is transformed by an interaction S• and a detector measures the overlap of the resulting state with a final state |out〉.• the transition probability P is given by the absolute square of the transitionamplitude AA in→out = 〈out S in〉P in→out = |A in→out | 2• if the initial and final states |in〉 and |out〉 are not pure states, the correspondingtransition probabilities must be added (e. g. spins and flavors) or integrated(e. g. angles, energies and momenta)Task(s):1. describe |in〉 and |out〉:• pure states: completetely polarized electrons, muons, photons⇒ Dirac equation, Klein-Gordon equation, &c.• mixtures: protons, partially polarized or unpolarized electrons, muons,photons, . . .2. compute S (i. e. the part of S that contributes to 〈out S in〉)• quantum electro dynamics (QED)• quantum chromo dynamics (QCD)• standard model (SM)• “new physics”, “beyond the SM” (BSM)⇒ Feynman rules3. square A in→out and integrate P in→out• Monte Carlo1(1a)(1b)
Page 1: Feynman Diagrams For PedestriansTho
Page 5 and 6: • a Lorentz transformation Λ mus
Page 7 and 8: 2.1 Klein-Gordon Equation(i∂ 0 )
Page 9 and 10: ∴ the formalism is consistent, if
Page 11 and 12: • we can verify the anti commutat
Page 13 and 14: • from which we can construct sol
Page 15 and 16: • invariant overlap from conserve
Page 17 and 18: k = (k 0 ; |⃗k| sin θ cos φ, |
Page 19 and 20: • that can be solved recursively
Page 21 and 22: ∵ intermediate states violate ene
Page 23 and 24: • the following Feynman rules pro
Page 25 and 26: • momentum conservation:s + t + u
Page 27 and 28: 4.2 Trace Techniques• consider a
Page 29 and 30: • traces9: trace4, 1;10: trace4,
Page 31 and 32: • the interference terms are more
Page 33 and 34: 5 QCD5.1 Feynman Rules∵ full acco
Page 35 and 36: with the “hadronic tensor”H µ
Page 37 and 38: 6 Standard Model(C, T) Y Q = T 3 +
Page 39 and 40: • Yukawa couplingspp, µp, µfZ 0
Page 41 and 42: 7 SolutionsSolution 1.∂ µ e −i
Page 43 and 44: Solution 6.ū k (p)γ µ u l (p) =
Page 45 and 46: Solution 14.L µν = tr [(/p + m)γ
Page 47 and 48: []iT 1 = ū(q 1 )(igT a /ɛ ∗ i(k
Page 49 and 50: • the Higgs mass shell follows fr

Feynman Diagrams For Pedestrians - Herbstschule Maria Laach

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