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 Quote by Sangoku Hi.. in what sense do you intrdouce the cut-off inside the action $$\int_{|p| \le \Lambda} \mathcal L (\phi, \partial _{\mu} \phi )$$ then all the quantities mass $$m(\Lambda)$$ charge $$q(\Lambda)$$ and Green function (every order 'n') $$G(x,x',\Lambda)$$ will depend on the value of cut-off, and are well defined whereas this cut-off is finite now what else can be done ??.. could we consider this cut-off $$\Lambda$$ to be some kind of 'physical' field (or have at least a physical meaning, or can we make this finite measuring 'm' 'q' or similar
I am not sure I understand your question but the cutoff represents the energy scale at which new physics becomes important.
Consider for example the Fermi model of the weak interaction. It`s an effective theory which can be used as long as the energy of the reaction is below the mass of the W boson. So you could construct an effective theory and integrate up to the mass of the W and renormalize and you would get a well defined expansion of any observable. but of the energy gets close to the mass of the W, the expansion breaks down because an infinite number of terms would have to be taken into account, signaling the need to use a mre fundamental theory.

hope this helps

Patrick