A query on the (old) motivation for renormalizable theories

[metroplex021]

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!

 

[tiny-tim]

hi metroplex021!

… 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. …

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

 

[metroplex021]

hi metroplex021!


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

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!

 

[tiny-tim]

hi metroplex021!

… 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!

sorry, I'm not understanding your terminology

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

but what do you mean by an "n-point function"?

 

[Avodyne]

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.

 

[metroplex021]

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!

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.