And so we really have h p 3 p n p 1 p 2 i amputated s

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and so we really have h p 3 · · · p n | p 1 p 2 i = (Amputated) × s Z R 2 Z OS 2 n . This is the renormalized version of the LSZ reduction formula, and it is cleaner in some sense. 24.2 Renormalized perturbation theory We have the bare Lagrangian L = - 1 4 ( μ A 0 ν - ν A 0 μ ) 2 + ¯ ψ 0 ( i/ - e 0 /A 0 - m 0 ) ψ 0 . Now we renormalize ψ 0 = Z 2 ψ R and A 0 μ = Z 3 A R μ with e 0 = Z e e R and m 0 = Z m m R . Then L = - 1 4 Z 3 ( μ A R ν - ν A R μ ) 2 + iZ 2 ¯ ψ R / ∂ψ R - Z 2 Z m m R ¯ ψψ - e R Z e Z 2 p Z 3 ¯ ψ R /A R ψ R . Now we define Z e Z 2 p Z 3 = Z 1 = 1 + δ 1 , Z 2 = 1 + δ 2 , Z 3 = 1 + δ 3 , Z m = 1 + δ m . We have that δ are formally of order at least 2 in e R even if they are infinite. So we can do perturbation theory in e R and then write L = - 1 4 F 2 μν + i ¯ ψ/ ∂ψ - m R ¯ ψψ - e R ¯ ψ /Aψ - 1 4 δ 3 F 2 μν + 2 ¯ ψ 2 / ∂ψ - ( δ m + δ 2 + δ m δ 2 ) m R ¯ ψψ - e R δ 1 ¯ ψ /Aψ where all these fields are now renormalized. Then we can read off the Feynman rules for these graphs, and for instance, = i ( / 2 - ( δ m + δ 2 + δ 2 δ m ) m R ) . We can think of this as these loops contracted to a point. We now want to choose δ so that we cancel out all the infinite parts. Now let us expand the electron self energy for instance. The we get i / p - m R + i / p - m R Σ 2 ( / p ) i / p - m R + i / p - m R ( i / 2 - ( δ m + δ 2 ) m R ) i / p - m R .
Physics 253a Notes 80 Similarly, we see that the photon self energy is - ie 2 R ( p 2 g μν - p μ p ν 2 ( p 2 ) - i ( p 2 g μν - p μ p ν ) δ 3 , Π 2 = 1 2 π Z 1 0 dxx (1 - x ) 2 + log ˜ μ 2 m 2 R - p 2 x (1 - x ) . To see the on-shell renormalization condition, we need to look at the sum of all the loop corrections and get ig μν p 2 (1 + Π( p 2 )) , Π( p 2 ) = e 2 R Π 2 ( p 2 ) + δ 3 . This has to look like ig μν /p 2 , so we get δ OS 3 = - e 2 R 6 π 2 1 - e 2 R 12 π 2 log ˜ μ 2 m 2 R . So we have dealt with all graph with two external legs. Now let us go to three legs. We have G 3 = h Ω | T { ¯ ψA μ ψ }| Ω i . Here, we can look at the 1PI graphs and write Γ μ ( p 2 ) = F 1 ( p 2 ) γ μ + i σ μν 2 m p ν F 2 ( p 2 ) , F 1 ( p 2 ) = 1 + e 2 R ( · · · ) , F 2 ( p 2 ) = 0 + α π + O ( p 2 ) . The loop graph contributing to this is just the graph we used to correct g - 2, and then we see that we get F 1 ( p 2 ) = 1 - 2 ie 2 R Z d 3 (1 - x - y - z ) Z d 4 k (2 π ) 4 k 2 - 2(1 - x )(1 - y ) p 2 + · · · [ k 2 - m 2 R (1 - x ) 2 · · · ] . As p 0, we should get the regular QED interaction, so we should have the δ 1 correction to be cancel out F 1 , that is, δ 1 = - F (2) 1 (0) = - α 2 π 1 + 1 2 log ˜ μ 2 m 2 R + 2 + log m 2 γ m 2 R . In fact, this is equal to δ 2 and so Z 1 = Z 2 . The reason for this is essentially charge conservation.
Physics 253a Notes 81 25 November 29, 2018 Today we will prove that QED is renormalizable. We introduced these renor- malizations for the photon self-interaction, the electron self-interaction, and the interaction term. Then we had L = - 1 4 Z 3 F 2 μν + iZ 2 ¯ ψ/ ∂ψ - e R Z 1 ¯ ψ /Aψ - m R Z 2 Z m ¯ ψψ. Then we had renormalization conditions that say that things look like tree-level in long distances. This fixes the 1-loop terms.
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