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A query on the (old) motivation for renormalizable theories 
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#1
Jul1311, 04:51 AM

P: 121

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 npoint 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 4point 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 nonpredictive? Any help much appreciated! 


#2
Jul1311, 06:24 AM

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hi metroplex021!
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 


#3
Jul1311, 10:32 AM

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#4
Jul1311, 05:21 PM

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A query on the (old) motivation for renormalizable theories
hi metroplex021!
by "external legs", i assume you mean eg 2>2 has 4 external legs? but what do you mean by an "npoint function"? 


#5
Jul1311, 09:33 PM

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An npoint vertex function is a sum of oneparticleirreducible 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 22 scattering, you need the 4point vertex function. But when you compute it, at high enough orders, you will have subdiagrams that involve npoint vertices for arbitrarily high n. And these have to be renormalized, so you will need the corresponding parameters. 


#6
Jul1411, 04:58 AM

P: 121

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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