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Homework Help: Chiral Anomaly (calculation)

  1. Jun 19, 2012 #1
    I am following Pierre Ramond's book "Field Theory - A Modern Primer", chapter 8, section 8.9, "Anomalies". My question starts in what follows from p.304.

    1. The problem statement, all variables and given/known data

    He draws the triangular diagram with one axial vector at one of the vertices and, using the standard Feynman rules, writes down the (superficially divergent) integral for the scattering amplitude of this process, eq. (8.9.61). He uses dimensional regularization, with dimension d=2ω.

    With some work on the integral and trace, we find that the integral has some pieces which, separately, diverge as ω→2 and other pieces which are immediately finite. But, gathering the divergent pieces together, we end up with another finite result.

    Now Pierre Ramond goes on to take the divergence of this result and basically says that only the divergent pieces (which are finite when taken as a whole) will contribute to the divergence (giving a finite nonvanishing result, the anomaly).

    My question is: is there some argument that justifies this statement, that the (superficially) finite part of the integral just doesn't contribute to the divergence (and therefore doesn't contribute to the anomaly)?

    2. Relevant equations

    3. The attempt at a solution

    I tried some "brute force" calculation: I gathered all (superficially) finite pieces and took the divergence of it. I did the calculation by myself, not following Ramond's one, and found that the divergence of the whole didn't vanish, so I probably made some mistake during the evaluation --- I shall repeat then following Ramond's results.

    But, afterall, what I'm really looking for is not just a "computational answer"... I'm more interested in a argument for it. For example, following the derivation of other books, we discover that the divergence of this diagram would vanish at first glance, after a naive shift of the momenta integration variable, but this turns out to be wrong because the integral is (superficially) divergent so we are not allowed to do this naive shifts: they actually generate relevant (nonvanishing) surface terms, which end up giving rise to the anomaly. Following this approach, I'm thinking if we could somehow justify on these grounds whether or not only the (superficially) divergent part of the diagram contributes to the anomaly.
    Last edited: Jun 19, 2012
  2. jcsd
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