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I am struggling to understand the concept of resummation of large logarithms in QFT; from what I learnt so far the problem relies on the fact that if a full theory defined in the UV contains much different energy scales, perturbation theory becomes useless because loop correction are usually of the form $\sim \log(\mu/\mu')$. Working with EFT should apparently simplify this problem, but I don't understand why.

In particular I don't get why the RG equation can help me to resum (I don't even understand what we mean by resummation, is it a way to get rid of them?) those large logs $\ldots$. Could you please make an example?

For example, in QED one can compute the 1-loop correction to the photon propagator and would find that (after the renormalization) $$\Pi(p^{2})\propto \log(p^{2}/\mu^{2})$$ so we have a large log correction if the scales are much different. As a consequence, one can show that $$\alpha(\mu)=\frac{\alpha_{\text{EM}}}{1-\frac{1}{3\pi}\alpha_{\text{EM}}\log(\mu^{2}/m^{2})}$$ Is this what we mean by resummation? As I understand if the log is large we have no problem our result doesn't get worse. Is it correct to say that?

Also, I do not understand why the EFT should be better at doing that. In my notes I've written that "only with EFT we can exploit the RG equation to promote the 1-loop result at any order in perturbation theory", but it makes no sense to me.

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I don't even understand what do we mean with resumming

Resummation literally means re-summation.

The one-loop $$ \Pi(p^{2})\propto log(p^{2}/\mu^{2}) $$ is the first non-constant term if we expand $\alpha(\mu)$: $$ \alpha(\mu)=\frac{\alpha_{EM}}{1-\frac{1}{3\pi}\alpha_{EM}log(\mu^{2}/m^{2})} \\ =1 + \frac{1}{3\pi}\alpha_{EM}^2log(\mu^{2}/m^{2}) + \dots $$

If you re-sum all the omitted $\dots$ terms, you get the full $\alpha(\mu)$.

Note that RG equation is not the only way to arrive at resummation. There are myriads of other ways to arrive at the same resummed formula. Just to name a few alternatives to RG:

  • Truncated Schwinger–Dyson equation
  • Large N approximation
  • Random-phase approximation
  • Bethe-Salpeter T-matrix approximation

All of the above alternative methodologies involve re-summing a (geometric) series of loop diagrams. And actually resummation has a very long history: Landau found the Landau pole by resumming the geometric series by brute force, a lot earlier than the invention of RG equation.

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  • $\begingroup$ Ok, but now I still miss the part where EFT should become more powerful than the complete theory. Do you know some good reference? $\endgroup$
    – Filippo
    Commented Jul 4 at 7:23