we have in Feynman gauge V VI i 3 ie 4 u 3 γ μ u 1 i� μν t ρ s p 4 σ s p 2 Z d

We have in feynman gauge v vi i 3 ie 4 u 3 γ μ u 1

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), we have (in Feynman gauge)V + VI=-i3(-ie)4u3γμu1-μνtρ*s0(p4)σs(p2)×Zd4k(2π)4Tr[γν(6k+6q+m)γρ(6k-6p2+m)γσ(6k+m)]((k+q)2-m2)((k-p2)2-m2)(k2-m2)+Tr[γν(-6k-6q+m)γρ(-6k+6p2+m)γσ(-6k+m)]((k+q)2-m2)((k-p2)2-m2)(k2-m2)whereq=p1-p3. The integral is a three-point functionhAν(p1-p3)Aρ(p2)Aσ(-p4)iwith the external propagators removed. It should vanish identically, and it’s easyto see why: Anything in the numerator that’s linear or cubic in momenta cancelsbetween the two terms. However, anything in the numerator that’s constant orquadratic in momenta has an odd number ofγ-matrices, and thus vanishes underthe trace.(c) We don’t have to do very much work here. The UV divergence in graph I comesfrom the wave-function renormalization subgraph on the internal electron line.But this is exactly the same graph that gives us the two-point counterterm, sothe counterterm exactly cancels that divergence.Similarly, graphs II and III have UV divergences from subgraphs that are similarto what we compute to get the vertex counterterm in QED. The only differenceis that the momenta flowing into the subgraph aren’t all on-shell. However, thevertex graph looks likeZd4k16k16k1k2It’s dimensionless and divergences are always polynomial in the momenta. Con-sequently, the divergence must be independent of the momenta flowing into thevertex graph, and it doesn’t care whether the incoming momenta are on or off-shell.Thus, the vertex counterterm suffices to cancel these divergences too.
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Phys 253a5Graphs V and VI vanish by Furry’s theorem as we just proved, and graph IVlooks likeZd4k16k16k16k1k2so is UV finite.(d) If the three point amplitude of photons did not vanish by symmetry we wouldhave needed a counterterm in our QED lagrangian which rendered it finite. Thiswould then take care of the divergence when it appeared as a subgraph, as in(b).3.(a) We have a few choices for what we call the “effective Yukawa” at the scalep,depending on what physical process we’re measuring. One possibility is just topick the coefficient ofiuuin the 1PI 3-point function. That’s what we’ll do here.
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  • SCHWARTZ
  • Quantum Field Theory, correlation function, momenta, charge conjugation invariance, counterterm

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