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Relation between dimensional regularization and high-energy modes

  1. Dec 19, 2015 #1
    I state that I am a beginner in QFT, but it seems to me that the methods to regularize the integrals of the perturbation series before renormalize serve to cut off the high-energy modes that are responsable for the UV divergences. This ( the cut off of high-energy modes ) nevertheless is not so obvius with regard to the dimensional regulaization.I would therefore understand the link that there is between the dimensional regularization and high-energy modes. Thank you
     
  2. jcsd
  3. Dec 21, 2015 #2

    vanhees71

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    There is none! Dimensional regularization is just a way to make the diverging integrals in perturburbation theory finite in such a way that you can subtract appropriate counter terms to the loop contributions to the proper vertex functions order by order perturbation theory and then take the physical limit, leading to a finite result within the chosen renormalization scheme. In the case of dimensional regularization you use the observation that many integrals become finite in less than four space-time dimensions (here and in the following I'm only talking about UV divergences, IR divergences in theories with massless particles are a different story). Thus you use the space-time dimension ##d## as a parameter and define a analytic continuation of the standard Feynman integrals as a function of ##d##. Then you Laurent expand around ##d=4##. The divergences of the integrals are now parametrized as poles ##1/(d-4)^n##, and after subtracting all subdivergences with the lower-order counter terms, you are left with a polynomial in the external momenta times a pole at ##d=4## which you can subtract with a local counterterm and then take set ##d=4## to obtain the finite result (that's the minimal subtraction scheme).

    The advantage of dimensional regularization compared to many other regularization procedures is that a lot of symmetries are preserved, particularly the Poinacare (or Euclidean if you work after Wick rotation in Euclidean QFT) symmetry and many (non-chiral) gauge symmetries. The disadvantage is that it is not so intuitive how (a) the renormalization scale comes into the game (it's introduced such as to keep the dimensions of the couplings the same as in four space-time dimensions, e.g., dimensionless for the usual gauge couplings in renormalizable gauge theories like QED or QCD) and (b) that the subtraction is always possible with local counter terms (in case of renormalizable theories of the same form as the terms in the Lagrangian you started with). To understand these issues better, it's worth to understand BPHZ renormalization, which works without any regularization in the beginning but just does the subtractions directly in the integrands of the divergent Feynman integrals, working systematically by subtracting the subdivergences first (the very elegant final description is given by Zimmermann's forest formula). For an introduction (mostly using the example of simple ##\phi^4## theory), see my qft lecture notes:

    http://fias.uni-frankfurt.de/~hees/publ/lect.pdf
     
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