By Forshaw J.
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Extra info for An introduction to QED and QCD
4) To understand how this cancellation of divergences happened we can study the convergence properties of loop diagrams (although we shall not evaluate them). 2. These both contain a loop with one photon propagator, behaving like 1/k 2 at large momentum k, and two electron propagators, each behaving like 1/k. 2). 3) arises because there is a divergence associated with the coupling of each electron in the scattering process). 2). 2? Now we have two photon and two electron propagators, leading to d 4k I∼ .
4 shows a graph where the divergence comes from the primitive divergent subgraph inside the dashed box. Furthermore, the primitive divergences are always of a type that would be generated by a term in the initial Lagrangian with a divergent coefficient. 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.
In terms of the renormalised fields 1 L = − Z3 FR µν FRµν + iZ2 ψ R ∂/ψR − Z1 eˆψ R A / R ψR − Zm Z2 mψ ˆ R ψR . 4 Writing each Z as Z = 1 + δZ, re-express the Lagrangian one more time as 1 L = − FR µν FRµν + iψ R ∂/ψR − eˆψ R A / R ψR − mψ ˆ R ψR + (δZ terms). 4 Now it looks like the old Lagrangian, but written in terms of the renormalised fields, with the addition of the δZ counterterms. Now when you calculate, the counterterms give you new vertices to include in your diagrams. The divergences contained in the counterterms cancel the infinities produced by the loop integrations, leaving a finite answer.