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Common wisdom is that for a QFT to be renormalizable it must be invariant under a symmetry transformation. Why does renormalization need an unbroken symmetry? Which is the first publication that proves it?

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This is a little vague. The result is that certain gauge theories are renormalizable. (Some random user of this site came up with this once.) Nevertheless, the electroweak theory is part of the standard model, renormalizable and broken. –  NikolajK Oct 31 '12 at 10:20
    
Technically it's not broken. The physically observable particle states are gauge invariant combinations of the background higgs & the bare particle states, neither of which is separately gauge invariant. I think there's a theorem to the effect that gauge invariance can't be broken. I think the "common wisdom" has to do with the fact that you generally need some symmetry in order to prevent quadratic - and therefore fine tuned - renormalizations. phi^4 theory is perturbatively renormalizable but has no symmetries. But it has quadratic mass renormalization. –  Foster Boondoggle Oct 31 '12 at 17:29
    
Actually I know that the opposite is true, in sense we can't renormlize gauge invariant theories without sacrificing with symmetry, thus any renormlized gauge invariant theory should has some symmetry break. –  TMS Oct 31 '12 at 21:13
    
I think the 'common wisdom' refers to 't Hooft proof of Yang-Mills theories beeing renormalizable link –  Somebody Nov 2 '12 at 12:11
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