Graduate Quantum Field Theory vs Effective Field Theory

[Luca_Mantani]

Hi everyone,
I'm approaching the study of EFT but I'm facing some problems. While in QFT usually we want renormalizable theories, in EFT we don't want this costraint anymore and this opens up space for a lot more terms in the Lagrangian.

My problem is that when we want to calculate amplitudes, if we want to go beyond the leading order, we have to deal with loops that, as usually, diverge and we have to renormalize adding counterterms. So, what's the point if we still have to renormalize? Why are they called non-renormalizable if we still renormalize the theory?

Thanks for the help,
Luca

 

[Dr.AbeNikIanEdL]

Well, I think basically any theory can be made finite by just adding enough counterterms, in the worst case you will need infinitely many. When one says that a theory is renormalizabel one usually means that it becomes finite to all orders by adding only a finite number of terms. In the case of effective field theories calculations only become finite up to some fixed order by adding a finite number of terms. So every time you want to go to a higher order you have to add more terms and more parameters, and still the next order will diverge.

 

[Luca_Mantani]

Oh ok, thanks! I was thinking that way too more or less, but I wasn't sure and I didn't find a confirmation. So renormalization is a tool that is still useful, even for theory that are not renormalizable!

 

[RGevo]

Definitely. Effective theories and not dissimilar to normal theories like qcd or whatever.

The higher order you can calculate in the EFT the better, since this gives you more accuracy for any predictions.

Ultimately, people struggle beyond NNLO in qcd, so you won't do better in an EFT framework which extends qcd for example. Therefore, although in principle the theory requires more and more counter terms for new operators etc., you can't even calculate far enough to require them.

So in the end, calculating to one loop in an EFT (like a dimension 6 extension of the SM) gives you higher precision, provides you sensitivity to correlations between operators which appear beyond tree level (through operator mixing), and can be quite fun figuring out how the one loop calculations actually work in the EFT.

 

[George Jones]

Luca_Mantani, for perspective and examples, you might want to look at chapters 21 - 23 of the book "Quantum Field Theory and the Standard Model" by Schwartz,
https://www.amazon.com/dp/1107034736/?tag=pfamazon01-20

 

[vanhees71]

Also note that in effective field theories you usually do an low-energy/momentum expansion, i.e., you have a "large scale" in the problem, and the small expansion parameter are energies/momenta compared to that large scale. E.g., in chiral perturbation theory the large scale is $4 \pi f_{\pi} \simeq 1 \; \mathrm{GeV}$. Usually effective theories are not Dyson-renormalizable but based on symmetries to constrain the possible terms in the Lagrangian. It can be shown that the theory can be renormalized at any order of the low-energy expansion, introducing more and more "low-energy constants" with any order, corresponding to the allowed terms in the Lagrangian.