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A query on the (old) motivation for renormalizable theories

  1. Jul 13, 2011 #1
    I've just realized I don't understand something pretty fundamental about the need to renormalize. Popular wisdom has it (or had it - forget the shift towards an effective framework) that theories that were not renormalizable had no predictive power, on account of the fact each n-point vertex function in such theories need to be renormalized anew, requiring new parameters to be measured at each n (see, e.g., Maggiore p139).

    But can't one say the following: say I am interested in studying only 2->2 interactions. Then presumably I only need to renormalize the 2, 3 and 4-point functions in order to derive predictions for these sorts of interactions. The infinitely many parameters apparently needed for a renormalizable theory (and once again, forget about EFTs) would only arise in the case that we study n->m particle relations in the limit that n & m go to infinity, which we never do. So why *were* renormalizable theories regarded as non-predictive?

    Any help much appreciated!
  2. jcsd
  3. Jul 13, 2011 #2


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    hi metroplex021! :smile:
    no, the number of input and output particles is irrelevant

    even with 2->2, there are infinitely many terms in the Dyson expansion …

    forget Feynman diagrams, it's those infinitely many terms that need to have a finite sum :wink:
  4. Jul 13, 2011 #3
    Well, I think that's how I was thinking before, but isn't the Dyson series written out for a single n-point function? (So that if you were ignoring interactions with less than n external legs, you wouldn't need to worry about that series - or indeed any series for any n'-point vertex function with n'>n?) Thanks!
  5. Jul 13, 2011 #4


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    hi metroplex021! :smile:
    sorry, i'm not understanding your terminology :redface:

    by "external legs", i assume you mean eg 2->2 has 4 external legs?

    but what do you mean by an "n-point function"? :confused:
  6. Jul 13, 2011 #5


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    An n-point vertex function is a sum of one-particle-irreducible diagrams with external propagators removed. (There is also a nonperturbative definition in terms of Legendre transforms of the functional integral with a source, but I don't remember it precisely enough to quote it.)

    For 2-2 scattering, you need the 4-point vertex function. But when you compute it, at high enough orders, you will have sub-diagrams that involve n-point vertices for arbitrarily high n. And these have to be renormalized, so you will need the corresponding parameters.
  7. Jul 14, 2011 #6
    That's awesome - thanks very much. I had a suspicion that was the case but have only ever worked at such a miniscule order I wasn't sure if it was the case. Thanks mate!

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