An introduction to QED and QCD by Forshaw J.

By Forshaw J.

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We have already met the Dirac spinors u and v. I will say more about the photon polarisation vector ǫ when we need to use it. 4 Electron–Muon Scattering To lowest order in the electromagnetic coupling, just one diagram contributes to this process. 1. The amplitude obtained by applying the Feynman rules to this diagram is iMf i = (−ie) u(pc )γ µ u(pa ) −igµν (−ie) u(pd )γ ν u(pb ). 44) Note that, for clarity, I have dropped the spin label on the spinors, I’ll restore it when I need to. In constructing this amplitude we have followed the fermion lines backwards with respect to fermion flow when working out the order of matrix multiplication (which makes sense if you think of an unbarred spinor as a column vector and a barred spinor as a row vector and remember that the amplitude carries no spinor indices).

Hence by rescaling the fields, masses and couplings in the original Lagrangian we can make all physical quantities finite (and independent of the exact details of the adjustment such as how we regulate the divergent integrals). This is what we mean by renormalisability. e. renormalised) parameters which have bare (unobservable) counterparts sitting in the original Lagrangian density. This means that a renormalisable theory has real predictive power. This should be made clearer by an example. Consider calculating the vertex correction in qed to one loop, µ ↓q = u(p′ ) Aγ µ + Bσ µν qν + Cq 2 γ µ + · · · u(p).

1 Lowest order Feynman diagram for electron–muon scattering. are what is left behind from the expansions of fields in terms of annihilation and creation operators, after the operators have all been (anti-)commuted until they annihilate the vacuum. The spinor indices in the Feynman rules are such that matrix multiplication is performed in the opposite order to that defining the flow of fermion number. 1. This will become clear in the examples which follow. We have already met the Dirac spinors u and v.

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