By Paul Roman

ISBN-10: 0201064952

ISBN-13: 9780201064957

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**Example text**

An alternative is to use dimensional regularization for the infrared divergences as well as the ultraviolet ones. As discussed in section 25, there are no soft-particle infrared divergences for d > 4 (and no colinear divergences at all in quantum electrodynamics with massive charged particles). In practice, infrared-divergent integrals are finite away from even-integer dimensions, just like ultraviolet-divergent integrals. Thus we simply keep d = 4 − ε all the way through to the very end, taking the ε → 0 limit only after summing over cross sections with extra undetectable photons, all computed in 4 − ε dimensions.

2m (271) where the ellipses stand for terms with more derivatives. Applied to Γ, the usual procedure for extracting the Feynman rules from an action yield a vertex factor that corresponds to eq. (269) with q 2 = 0. To see this, recall that an incoming photon translates into a factor of Aµ ∼ ε∗µ eiqx , and therefore of Fµν ∼ i(qµ ε∗ν −qν ε∗µ )eiqx ; the two terms in Fµν cancel the extra factor of one half in the second term in eq. (271). Now we will see what eq. (271) predicts for the magnetic moment of the electron.

We have iΣ(/ p) = (iZ1 e)2 1 i d4ℓ ˜ p + /ℓ)γ µ ∆ ˜ µν (ℓ) γ ν S(/ (2π)4 2 − i(Z2 −1)/ p − i(Zm −1)m + O(e4 ) . (223) It is simplest to work in Feynman gauge, where we take ˜ µν (ℓ) = ∆ ℓ2 gµν ; + λ2 − iǫ (224) here we have included the fictitious photon mass λ as an infrared cutoff. We now apply the usual bag of tricks to get iΣ(/ p) = e2 µ ˜ε 1 0 dx ddq N d 2 (2π) (q + D)2 − i(Z2 −1)/ p − i(Zm −1)m + O(e4 ) , (225) where q = ℓ + xp and D = x(1−x)p2 + xm2 + (1−x)λ2 , (226) N = γµ (−/ p − /ℓ + m)γ µ = −(d−2)(/ p + /ℓ) − dm = −(d−2)[/q + (1−x)/ p] − dm , 55 (227) where we have used (from section 47) γµ γ µ = −d and γµ p/γ µ = (d−2)/ p.

### Advanced Quantum Theory by Paul Roman

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