Wilsonian EFTs and regularization

[metroplex021]

Hi folks - I have a couple of questions about EFTs that are driving me crazy.

(1 ) Consider first of all a Wilsonian effective Lagrangian - one in which particles of mass >M have been integrated out from a 'full' Lagrangian leaving a string of non-renormalizable interactions amongst the light particles. If we want to make predictions using this theory, we will need to use a renormalization scheme. In so doing, are we
(a) compelled to use a momentum-space regularization, with the cut-off placed at M;
(b) are we forbidden to take this cut-off to infinity, thus committed to viewing spacetime as a lattice?

(I am wondering this because I'm wondering if you can use the Wilson technique for making effective Lagrangians and still have the resulting EFT defined on a continuum. I'm also wondering this because the two kinds of cut off used in the Wilsonian picture - at least as presented in Peskin and Schroeder - seem conceptually distinct: one is telling us which physical particles are going to be relevant at a given energy, the other cutting off the types of virtual particles that might contribute. But maybe these two roles are after all one and the same role.)

(2) Now for a question on dimensional regularization. With a momentum cut-off we can at least make sense of not taking the cut-off to infinity (putting the violence that it does to Poincare invariance and the structure of spacetime). Is there any sense in using dimensional regularization and not going to the d=4 limit at the end of the calculation? That is, do we have the option of not removing the 'cut-off' in this case?

Any thoughts or references would be most appreciated!

 

[TriTertButoxy]

I remember having a somewhat heated debated about this with my fellow colleagues not too long ago (I have not come to consensus with them). My answer is this: The sole purpose of a regulator is to extract the UV behavior of a theory needed to make the renormalization process a respectable mathematical procedure. In ALL cases, I claim that the regulator MUST BE REMOVE upon renormalization, regardless of the type of regulator (dim. reg. or cut-off) and whether the theory is effective or not.

In your example effective theory, in which particles of mass greater than M have been integrated out (but not any momenta of the remaining particles), you MUST still integrate over all remaining degrees of freedom (ie all momentum). No lattice business.

 

[TriTertButoxy]

There is one exception; if fields carrying momentum greater than scale K have been integrated out, then any future computations (ie a loop calculation) from the resulting effective theory should not include these DOFs as they have already been taken into account. Such a theory should have no UV divergence at all.

 

[metroplex021]

Thanks very much for your help. A difficulty here is that when many places I've looked at discuss the 'integrating out' procedure, they talk about " 'heavy', that is high momentum " fields. So if there is indeed a distinction to make here, it's rarely made. I think this is the most confusing topic I've ever read about!