Feynman Diagrams For Pedestrians - Herbstschule Maria Laach

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1.4 Units• from now on, we will use units which will give us numbers with natural orderof magnitude for quantum mechanics and relativistic kinematics̷h = c = 1 . (17)• velocities and actions are dimensionless and therefore[ ]1[energy] = [momentum] = [mass] = . (18)length• in particular, our <strong>Feynman</strong> rules, will later yield cross sections in units of [energy −2 ],e. g.• the relevant conversion factors are(TeV 2 nb = GeV 2 mb) and therefore2 Asymptotic Statesσ = 4πα23E 2 (19)̷hc = 197.327 053(59) MeV fm (20)(̷hc) 2 = 0.389 379 66(23) TeV 2 nb (21)σ =• described by wave equations, that are4πα 23(E/TeV) 2 0.39 nb (19′ )1. linear: superposition principle of quantum mechanics2. relativistic: matrix elements of observables must transform under rotationsand Lorentz boosts like scalars, four vectors, tensors, &c.3. and have the correct dispersion relation: E 2 = ⃗p 2 + m 2• objects of interest– spin-0 particles: not yet unambigously observed as an elementary particle,but ATLAS and CMS have strong evidence for a state with spin 0 or 2∗ one invariant component– spin-1/2 particles: leptons, quarks∗ at least two components: spinor under rotations– spin-1 particles: gauge bosons∗ massive three components (polarizations)∗ massless two components (polarizations)4
2.1 Klein-Gordon Equation(i∂ 0 ) 2 φ(x) =[(−i⃗∂) 2 + m 2 ]φ(x) (22)• is obviously a covariant wave equation, because(□ + m2 ) φ(x) = 0 (23)• fourier transformφ(x) =∫ d 4 p(2π) 4 e−ipx ˜φ(p) , i∂ µ φ(x) =∫ d 4 p(2π) 4 e−ipx p µ ˜φ(p) , etc (24)∴ algebraic equation (p 2 − m 2) ˜φ(p) = 0 (23 ′ )p 0“mass shell”:p 0 = + √ ⃗p 2 + m 2 ,p 2 = m 2 , p 0 0|⃗p|p 0 = − √ ⃗p 2 + m 2• correct relativistic dispersion relation E = + √ ⃗p 2 + m 2• but what about the other solution E = − √ ⃗p 2 + m 2 ?2.2 Free Spin-0 Particles• general solution of the Klein-Gordon equation∫ d 4 pφ(x) =(2π) 2πΘ(p 0)δ(p 2 − m 2 ) ( φ (+) (⃗p)e −ipx + φ (−) (⃗p)e ipx) (25)4∫ ∣d 3 ⃗p ∣∣∣∣p0 (=φ(2π) 3 2p 0=√(+) (⃗p)e −ipx + φ (−) (⃗p)e ipx) (26)⃗p 2 +m∫2= ˜dp ( φ (+) (⃗p)e −ipx + φ (−) (⃗p)e ipx) (27)• conserved current ∂ 0 j 0 (x) − ⃗∇⃗j(x) = ∂ µ j µ (x) = 0 out of two solutions φ 1 and φ 2of the Klein-Gordon equation with the same mass:j µ (x) = φ ∗ 1(x)i ←→ ∂ µ φ 2 (x) = φ ∗ 1(x)[i∂ µ φ 2 (x)] − [i∂ µ φ ∗ 1(x)]φ 2 (x) (28)5
Page 1 and 2: Feynman Diagrams For PedestriansTho
Page 3 and 4: 1 Introduction1.1 Scattering Amplit
Page 5: • a Lorentz transformation Λ mus
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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