Effective field theory (EFT) and Renormalizability

Question

Was trying to understand renormalizability in EFT. This is a little confusing especially the part of the misnomer. Can someone please explain this?

Text taken from Wikipedia:

"However, in an effective field theory, "renormalizability" is, strictly speaking, a misnomer. In a nonrenormalizable effective field theory, terms in the Lagrangian do multiply to infinity, but have coefficients suppressed by ever-more-extreme inverse powers of the energy cutoff. If the cutoff is a real, physical quantity—if, that is, the theory is only an effective description of physics up to some maximum energy or minimum distance scale—then these extra terms could represent real physical interactions. Assuming that the dimensionless constants in the theory do not get too large, one can group calculations by inverse powers of the cutoff, and extract approximate predictions to finite order in the cutoff that still have a finite number of free parameters. It can even be useful to renormalize these "nonrenormalizable" interactions."

Answer 1

The article assumes that the reader thinks that the point of renormalization is removing infinities. What it is saying is that that issue of renormalization is not simply about removing infinities but about how physics changes as scales change. This concept is much more general and can be applied to situations when there are no infinities in sight like in condensed matter systems.

Answer 2

So EFTs are quite allusive things that most people use to renormalize a theory, but they are so much more powerful than that. I mainly look at EFTs of gravity so I'll take it from that approach.

Wilson's approach to renormalisation relies on an EFT based approach. At a basic level one writes down the effective terms to some given order in the operators and then uses this to rescale the coupling constant in the full theory so that the full theory does not diverge at a given loop level. This is known as Top Down EFT.

The other case is Bottom-Up EFT in which we begin with some low energy theory and then (as your quote says) add higher order terms which represent corrections to the Low Energy Effective Field Theory (LEEFT). In the gravitational case, we usually take as our LEEFT the Einstein Hilbert action

$$S=M_p^2\int d^4x\sqrt{-g}R,$$

where $M_p$ is the Planck mass. Now this action is only valid for low energies, up to some mass scale $\approx M_p$. The dimension of the Einstein-Hilbert term $[R]=2$. That is why the Planck mass appears as a pre-factor, to fix up the dimensionality. But now let us go to a slightly higher energy given by $\Lambda=M_p/g_*$ where $g_*<1$ is some characteristic coupling. We could add on something like the Weyl term $C_{\mu\nu\rho\sigma}$ which is also dimension-2, but we can square it to give, $$S'=M_p^2\int d^4x\sqrt{-g}\bigg(R-\frac{c_1}{\Lambda^2}(C^{\mu\nu\rho\sigma})^2\bigg),$$ where we have had to suppress the Weyl-squared term. We can keep adding on an infinite number of higher-order operators with couplings $\{c_i\}$ suppressed by higher orders of $\Lambda$. This is an EFT from the bottom-up. In the limit of the infinite sum, and assuming we have some method to determine $\{c_i\}$, we havea theory of quantum gravity.

So we can either renormalise a theory, whose full theory we know, via this technique,

OR

we can start with a low energy theory and add higher order operators in order to explore higher energies.