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Let be a dimensional regularized integral

$$ \int d^{4-\epsilon}kF(k,m,s)= \frac{2}{\epsilon}+\frac{m^{2}}{3}(\gamma +log(4\pi)-\frac{1}{\epsilon}))$$

then formally if we elmiinate the divergent quantities we may have

$$ \int d^{4-\epsilon}kF(k,m,s)_{reg}= \frac{m^{2}}{3}(\gamma +log(4\pi)+log\mu) $$

here $m$ and $s$ are parameters and $ \mu $ and energy scale

but is it that enoguh is renormalization simply 'deleting' teh divergent quantities proportional to $ \frac{1}{\epsilon ^{k}} $ ?

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I'm not sure how you got your results, but renormalization isn't simply deleting the $\frac{1}{\epsilon}$ terms, we need to choose our counterterms such that they remove these divergences. – Will Jul 3 '13 at 19:44

Taking your integral as an example, i.e.

$$\int \mathrm{d}^{4-\epsilon}k \, \, F(k,m,s) = \frac{2}{\epsilon} + \frac{m^2}{3}\left( \gamma + \log (4\pi) -\frac{1}{\epsilon} \right)$$

Renormalization does not simply 'delete' the $1/\epsilon$ divergences. It is a well-defined procedure which expresses the amplitudes in terms of renormalized, measurable and physical quantities rather than the bare parameters (e.g. $e$ and $m$) which appear in the original Lagrangian. The amplitudes cannot be infinite, and hence if the procedure is executed correctly, it must inevitably remove divergences.

For additional resources regarding renormalization, I recommend:

  1. Peskin and Schroeder's Introduction to Quantum Field Theory which provides a thorough (and sufficiently explicit) introduction to renormalization and the general theory known as the renormalization group which further explores ideas such as scale-dependent quantities, e.g. coupling constant.
  2. Srednicki's Quantum Field Theory which also covers renormalization, and is available for free at Furthermore, the text provides some calculations in much more detail than Peskin and Schroeder, however as a whole I believe that the first resource is superior.
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