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In renormalized perturbation theory, it's unclear to me how exactly we add the necessary counter-terms. Do we:

  1. Draw all possible diagrams, including the diagrams of the counter-terms to some order from the Feynman rules automatically, compute all of them and thus cancel all divergences,

or

  1. First draw the ordinary diagrams, without the counter terms, compute them and then add counter-terms diagrams that should cancel the divergences that arose and use renormalization conditions to compute them?
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  • $\begingroup$ First, you need to have renormalized Lagrangian. I mean you need to rescale the fields and coupling constants. Then your Lagrangian will look like $L_{field}= L_{renormalized}+ L_{counterterms}$. And do not forget, Feynman diagrams applicable in momentum space. So you should consider the derivatives i.e, $\partial$, as momentum $p$. $\endgroup$ – aQuestion Jun 28 '15 at 9:47
  • $\begingroup$ First, you isolate the divergent pieces by looking at potentially divergent diagrams. For example, if you use dimensional regularization then typically the poles are inverse powers of $\epsilon$ where $\epsilon \rightarrow 0$ makes the spacetime dimension $D$ equal the actual spacetime dimension you're working in. Then, for each such divergent piece, you add a counter-term (with suitably embellished multiplicative prefactors) to the Lagrangian (or effective action) which cancels the divergence. $\endgroup$ – leastaction Jun 28 '15 at 15:26

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