Feynman Diagrams For Pedestrians - Herbstschule Maria Laach

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• NB: this is not what happens in the standard model, where the mass comessolely from the coupling to the Higgs sector and a gauge freedom remains.However (102) is in lowest order equivalent to the Higgs mechanism in unitaritygauge• NB: propagators and external states are universal and models differ only in theparticle content and interaction vertices!3.2 <strong>Feynman</strong> Rulesexternal spin-1/2 particles:incoming:outgoing:pp⇐⇒ · · · u(p)⇐⇒ ū(p) · · ·(103a)(103b)external spin-1/2 anti particles:incoming:outgoing:pp⇐⇒ ¯v(p) · · ·⇐⇒ · · · v(p)(103c)(103d)external spin-1 particles:incoming:outgoing:kk⇐⇒ ɛ µ (k)⇐⇒ ɛ ∗ µ(k)(103e)(103f)• internal particles and anti particles with sign of momentum relative to the arrow(i. e. charge) direction:spin-1/2:p⇐⇒i/p − m + iɛ(104a)spin-1 (m = 0):k⇐⇒ −ig µν + i(1 − ξ) k µk νk 2 +iɛk 2 + iɛ(104b)• finallyspin-0:p⇐⇒ip 2 − m 2 + iɛ(105)∵ the S-matrix always contains an uninteresting diagonal piece (no interaction) andthe global momentum conservation δ-distributionS = 1 + (2π) 4 δ 4 (p 1 + p 2 − q 1 − q 2 . . . − . . . q n )iT (106)∴ we can focus on T20
• the following <strong>Feynman</strong> rules produce an expression for iT:1. draw all diagrams using propagators and interaction vertices that connectthe initial state with the final state2. assign momenta and external wave functions accordingly3. use momentum conservation at each vertex to fix the momenta of the internallines4. follow each connected fermion line against the direction of the arrows andcollect wave functions, propagators and vertices along the way5. complete iT with the remaining wave functions, propagators and vertices6. add all diagrams with the relative signs such that that sum is anti symmetricunder the exchange of identical external (anti-)fermions (and symmetric forbosons)• in diagrams with closed loops not all momenta are fixed by momentum conservation,e. g.:p − kkpk• in this case, one must integrate over these loop momenta with∫ d 4 p(2π) 4 · · · (107)• unfortunately, there are infinitely many loop diagrams for each process• fortunately, each loop comes with additional powers of the coupling constants• in weakly interacting theories, we can expand the scattering amplitudes in thecouplings constants or the number of loops3.3 Cross Section• definition in terms of observablesσ (∆Φ) = R (∆Φ)jwhere (108)∆Φ = region of phase spaceσ (∆Φ) = cross section for scattering into ∆ΦR (∆Φ) = event rate in ∆Φj = incoming flux(109a)(109b)(109c)(109d)21
Page 1 and 2: Feynman Diagrams For PedestriansTho
Page 3 and 4: 1 Introduction1.1 Scattering Amplit
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: ∵ intermediate states violate ene
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

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