. α β γ δ ε ζ η θ ι κ λ μ Aμ ν(x) ξ ο π ρ ς σ τ υ φ χ ψ ω. Α Β Γ Δ Ε Ζ Η Θ Ι Κ Λ Μ Ν Ξ Ο Π Ρ Σ Τ Υ Φ Χ Ψ Ω. Friday April 1 ± ǁ

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1 . α β γ δ ε ζ η θ ι κ λ μ Aμ ν(x) ξ ο π ρ ς σ τ υ φ χ ψ ω. Α Β Γ Δ Ε Ζ Η Θ Ι Κ Λ Μ Ν Ξ Ο Π Ρ Σ Τ Υ Φ Χ Ψ Ω Friday April 1 ± ǁ 1

2 Chapter 5. Photons: Covariant Theory 5.1. The classical fields 5.2. Covariant quantization 5.3. The photon propagator Chapter 6. The S-Matrix Expansion 6.1. Natural Dimensions and Units 6.2. The S-matrix expansion 6.3. Wick s theorem Chapter 7. Feynman Diagrams and Rules in QED 7.1. Feynman diagrams in configuration space 7.2. Feynman diagrams in momentum space 7.3. Feynman rules for QED 7.4. Leptons Chapter 8. QED Processes in Lowest Order 8.1. The cross section 8.2. Spin sums 8.3. Photon polarization sums Examples Bremsstrahlung SECTION 7.2. FEYNMAN DIAGRAMS IN MOMENTUM SPACE First, let s consider an example: ee scattering; the Møller cross section. The lowest order term is 2nd order in the interaction, S (2) = (1 /2!) (ie) 2 d 4 x d 4 y < f T ψγ μ ψ A μ (x) ψγ ν ψa ν (y) i > where i > = e 1 ; e 2 > and f > = e 3 ; e 4 > 2

3 S (2) = (1 /2!) (ie) 2 d 4 x d 4 y < f T ψγ μ ψ A μ (x) ψγ ν ψa ν (y) i > Apply Wick s theorem and the coordinate space Feynman rules. Note: The 1 / n! from the exponential series, will always cancel n! permutations of the vertex positions { x 1, x 2, x 3,, x n }. This leads to a Feynman rule: (1 ) Draw all the topologically distinct diagrams with the specified external lines. For ee scattering there are two Feynman diagrams, Wick s theorem gives us 4 terms. However, S 22 = S 11 and S 21 = S 12 because we integrate over d 4 x and d 4 y. (E.g., in S 22 change the variables of integration from x,y to y,x ; then drop the primes; the result is S 11.) So, S = (S 11 +S 12 ) 2 ; the 2 will cancel the 1 / 2!. The corresponding S-matrix elements are 3

4 Transform S to momentum space. The integral d 4 y : D μν (x y) = (2π) 4 d 4 k ( g μν /k 2 ) e i k.(x y) For the case S t : the integral d 4 x gives Comment: That would be for infinite spacetime. Instead, we normalize plane waves in a finite volume Ω, so then the result should be The transformation from coordinate space to momentum space gave us some delta functions for 4-momentum conservation. This leads to another Feynman rule: (2) 4-momentum is conserved at every vertex. These are the same in the limit Ω, but... 4

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6 The Feynman rules in momentum space are rules for calculating the matrix element M. The S-matrix has some normalization factors that are not included in M. These normalization factors are (2 Ω E i ) 1/2 and (2 Ω E f ) 1/2 i f and (2m) 1 /2. e These factors are not part of M. The matrix element for ee (Møller) scattering M t = e 2 (u 3 γ μ u 1 ) (u 4 γ μ u 2 ) /t M u = e 2 (u 3 γ μ u 2 ) (u 4 γ μ u 1 ) /u M = M t + M u Note these other Feynman rules: (3) A spinor for every external electron and positron. (4) A polarization vector for every external photon. (5) A propagator S F (p) for every (internal) electron line. (6) A propagator D F μν (q) for every (internal) photon line. (7) A minus sign for exchanging 2 electrons. 6

7 To complete the calculation of the Møller cross section, we will need: M 2 ; and the average over initial spins and sum over final spins, i.e., ½ ½ [...]. λ 1 λ 2 λ 3 λ 4 Let s go ahead and calculate that now. Homework Problem X: Plot the Moller cross section. 7

8 The cross section. The transition probability is the square of the S- matrix element, P = S 2. The transition rate is the probability per unit time, w = P/(2T) (evolution from -T to T) The cross section is the rate divided by the incident flux, and the incident flux is Φ = density velocity = v rel / Ω ; (1 particle in the volume Ω ) Thus dσ = w / Φ = S 2 Ω 2T v rel 8