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Wilsonian EFTs and regularization

  1. May 19, 2010 #1
    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!
  2. jcsd
  3. May 19, 2010 #2
    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.
  4. May 19, 2010 #3
    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.
  5. May 20, 2010 #4
    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!
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