Friday, July 19, 2019

quantum field theory - Renormalizibility by power counting


When testing a theory for its renormalizability, in practice one always calculates the mass dimension of the coupling constants gi. If [gi]<0 for any i the theory is not renormalizable. I am wondering where this criterion/trick comes from? Is there an easy way to see that a coupling constant with negative mass dimension will yield a non-renormalizable theory?



Answer



Yes, suppose [g]=δ. By dimensional analysis only we can write that a loop diagram contributes gnd4kk4nδ

If δ=0, this diverges logarithmically, but can be re-normalized. If δ is less than zero, it diverges by simple power counting.


This is VERY informal. Technically, you should study the superficial degree of divergence of a diagram. But that's called superficial for a reason. So for the whole story I think you need Weinberg's theorem, which is a rule for telling exactly if a diagram diverges.


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